Maple Questions and Posts

These are Posts and Questions associated with the product, Maple

Is there or will there be a german version or is there a language pack available somewhere?

I was wondering if you can index arrays in maple with a loop. In MATLAB, you can index all rows and columns in the Matrix, A, with A(:,:), or a portion of the elements such as A(2:10,5:20) to index, or "loop" through these specific elements. In Maple, I am trying to index the following elements, as done in MATLAB with the command 

dIdQ(nmid-(imax-1)/2:nmid+(imax-1)/2,1) = -(omega_t(1:imax,1) - omega_1(nmid-imax*jmax-(imax-1)/2:nmid-imax*jmax+(imax-1)/2,1));

Is there a way to convert this matlab code to maple code?

Any help is very appreciated. Thank you!

r=a+b sin(x), this is the function I want to plot the graph in polar co-ordinates but how to manage those arbitrary constants "a and b" ?

Hi everyone, I need someone’s expertise with the Maple Soft Application, I am currently into my 4th day of my Maple Soft experience and also returning to School after many years. Can someone help me with the following codes to solve step by step Differential Equation 1 below;

Please help me with following codes below

  1. Boundary Condition 1
  2. Boundary Condition 2
  3. General Solution
  4. The integration Part of the Equation
  5. Solving the Differential Equation  

Equation 1.

Determine the equilibrium temperature for a one dimensional rod with a constant thermal properties with the following source and boundary conditions.

  1. Q = 0, U(0) = 10, u(L) = 20
  2. Q/K0 = x, u(0) = 0, u(L) = 10

 

I understand how to obtain the solution to the equation above,

(a) Equilibrium satisfies

                                                U’’(x) = 0,

Whose general solution is u = c1 + c2x.

The boundary condition u(0) = 0 implies c1 = 0 and u(L) = T implies c2 = T/L so that u = T x/L.

(f) In equilibrium, u satisfies

U’’(x) = −Q/K0 = −x^2,

Whose general solution (by integrating twice) is

u = −x ^4 /12 + c1 + c2x.

The boundary condition u(0) = T yields c1 = T, while u’(L) = 0 yields c2 = L^3/3.

Thus u = −x ^4 /12 + L^3x/3 + T.

Integral Transforms (revamped) and PDEs

 

Integral transforms, implemented in Maple as the inttrans  package, are special integrals that appear frequently in mathematical-physics and that have remarkable properties. One of the main uses of integral transforms is for the computation of exact solutions to ordinary and partial differential equations with initial/boundary conditions. In Maple, that functionality is implemented in dsolve/inttrans  and in pdsolve/boundary conditions .

 

During the last months, we have been working heavily on several aspects of these integral transform functions and this post is about that. This is work in progress, in collaboration with Katherina von Bulow

 

The integral transforms are represented by the commands of the inttrans  package:

with(inttrans)

[addtable, fourier, fouriercos, fouriersin, hankel, hilbert, invfourier, invhilbert, invlaplace, invmellin, laplace, mellin, savetable, setup]

(1)

Three of these commands, addtable, savetable, and setup (this one is new, only present after installing the Physics Updates) are "administrative" commands while the others are computational representations for integrals. For example,

FunctionAdvisor(integral_form, fourier)

[fourier(a, b, z) = Int(a/exp(I*b*z), b = -infinity .. infinity), MathematicalFunctions:-`with no restrictions on `(a, b, z)]

(2)

FunctionAdvisor(integral_form, mellin)

[mellin(a, b, z) = Int(a*b^(z-1), b = 0 .. infinity), MathematicalFunctions:-`with no restrictions on `(a, b, z)]

(3)

For all the integral transform commands, the first argument is the integrand, the second one is the dummy integration variable of a definite integral and the third one is the evaluation point. (also called transform variable). The integral representation is also visible using the convert network

laplace(f(t), t, s); % = convert(%, Int)

laplace(f(t), t, s) = Int(f(t)*exp(-s*t), t = 0 .. infinity)

(4)

Having in mind the applications of these integral transforms to compute integrals and exact solutions to PDE with boundary conditions, five different aspects of these transforms received further development:

• 

Compute Derivatives: Yes or No

• 

Numerical Evaluation

• 

Two Hankel Transform Definitions

• 

More integral transform results

• 

Mellin and Hankel transform solutions for Partial Differential Equations with boundary conditions


The project includes having all these tranforms available at user level (not ready), say as FourierTransform for inttrans:-fourier, so that we don't need to input with(inttrans) anymore. Related to these changes we also intend to have Heaviside(0) not return undefined anymore, and return itself instead, unevaluated, so that one can set its value according to the problem/preferred convention (typically 0, 1/2 or 1) and have all the Maple library following that choice.

The material presented in the following sections is reproducible already in Maple 2019 by installing the latest Physics Updates (v.435 or higher),

Compute derivatives: Yes or No.

 

For historical reasons, previous implementations of these integral transform commands did not follow a standard paradigm of computer algebra: "Given a function f(x), the input diff(f(x), x) should return the derivative of f(x)". The implementation instead worked in the opposite direction: if you were to input the result of the derivative, you would receive the derivative representation. For example, to the input laplace(-t*f(t), t, s) you would receive d*laplace(f(t), t, s)/ds. This is particularly useful for the purpose of using integral transforms to solve differential equations but it is counter-intuitive and misleading; Maple knows the differentiation rule of these functions, but that rule was not evident anywhere. It was not clear how to compute the derivative (unless you knew the result in advance).

 

To solve this issue, a new command, setup, has been added to the package, so that you can set "whether or not" to compute derivatives, and the default has been changed to computederivatives = true while the old behavior is obtained only if you input setup(computederivatives = false). For example, after having installed the Physics Updates,

Physics:-Version()

`The "Physics Updates" version in the MapleCloud is 435 and is the same as the version installed in this computer, created 2019, October 1, 12:46 hours, found in the directory /Users/ecterrab/maple/toolbox/2019/Physics Updates/lib/`

(1.1)

the current settings can be queried via

setup(computederivatives)

computederivatives = true

(1.2)

and so differentiating returns the derivative computed

(%diff = diff)(laplace(f(t), t, s), s)

%diff(laplace(f(t), t, s), s) = -laplace(f(t)*t, t, s)

(1.3)

while changing this setting to work as in previous releases you have this computation reversed: you input the output (1.3) and you get the corresponding input

setup(computederivatives = false)

computederivatives = false

(1.4)

%diff(laplace(f(t), t, s), s) = -laplace(t*f(t), t, s)

%diff(laplace(f(t), t, s), s) = diff(laplace(f(t), t, s), s)

(1.5)

Reset the value of computederivatives

setup(computederivatives = true)

computederivatives = true

(1.6)

%diff(laplace(f(t), t, s), s) = -laplace(t*f(t), t, s)

%diff(laplace(f(t), t, s), s) = -laplace(f(t)*t, t, s)

(1.7)

In summary: by default, derivatives of integral transforms are now computed; if you need to work with these derivatives as in  previous releases, you can input setup(computederivatives = false). This setting can be changed any time you want within one and the same Maple session, and changing it does not have any impact on the performance of intsolve, dsolve and pdsolve to solve differential equations using integral transforms.

``

Numerical Evaluation

 

In previous releases, integral transforms had no numerical evaluation implemented. This is in the process of changing. So, for example, to numerically evaluate the inverse laplace transform ( invlaplace  command), three different algorithms have been implemented: Gaver-Stehfest, Talbot and Euler, following the presentation by Abate and Whitt, "Unified Framework for Numerically Inverting Laplace Transforms", INFORMS Journal on Computing 18(4), pp. 408–421, 2006.

 

For example, consider the exact solution to this partial differential equation subject to initial and boundary conditions

pde := diff(u(x, t), x) = 4*(diff(u(x, t), t, t))

iv := u(x, 0) = 0, u(0, t) = 1

 

Note that these two conditions are not entirely compatible: the solution returned cannot be valid for x = 0 and t = 0 simultaneously. However, a solution discarding that point does exist and is given by

sol := pdsolve([pde, iv])

u(x, t) = -invlaplace(exp(-(1/2)*s^(1/2)*t)/s, s, x)+1

(2.1)

Verifying the solution, one condition remains to be tested

pdetest(sol, [pde, iv])

[0, 0, -invlaplace(exp(-(1/2)*s^(1/2)*t)/s, s, 0)]

(2.2)

Since we now have numerical evaluation rules, we can test that what looks different from 0 in the above is actually 0.

zero := [0, 0, -invlaplace(exp(-(1/2)*s^(1/2)*t)/s, s, 0)][-1]

-invlaplace(exp(-(1/2)*s^(1/2)*t)/s, s, 0)

(2.3)

Add a small number to the initial value of t to skip the point t = 0

plot(zero, t = 0+10^(-10) .. 1)

 

The default method used is the method of Euler sums and the numerical evaluation is performed as usual using the evalf command. For example, consider

F := sin(sqrt(2*t))

 

The Laplace transform of F is given by

LT := laplace(F, t, s)

(1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2)

(2.4)

and the inverse Laplace transform of LT in inert form is

ILT := %invlaplace(LT, s, t)

%invlaplace((1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2), s, t)

(2.5)

At t = 1 we have

eval(ILT, t = 1)

%invlaplace((1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2), s, 1)

(2.6)

evalf(%invlaplace((1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2), s, 1))

.9877659460

(2.7)

This result is consistent with the one we get if we first compute the exact form of the inverse Laplace transform at t = 1:

%invlaplace((1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2), s, 1) = value(%invlaplace((1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2), s, 1))

%invlaplace((1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2), s, 1) = sin(2^(1/2))

(2.8)

evalf(%invlaplace((1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2), s, 1) = sin(2^(1/2)))

.9877659460 = .9877659459

(2.9)

In addition to the standard use of evalf to numerically evaluate inverse Laplace transforms, one can invoke each of the three different methods implemented using the MathematicalFunctions:-Evalf  command

with(MathematicalFunctions, Evalf)

[Evalf]

(2.10)

Evalf(%invlaplace((1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2), s, 1), method = Talbot)

.9877659460

(2.11)

MF:-Evalf(%invlaplace((1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2), s, 1), method = GaverStehfest)

.9877659460

(2.12)

MF:-Evalf(%invlaplace((1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2), s, 1), method = Euler)

.9877659460

(2.13)

Regarding the method we use by default: from a numerical experiment with varied problems we have concluded that our implementation of the Euler (sums) method is faster and more accurate than the other two.

 

Two Hankel transform definitions

 


In previous Maple releases, the definition of the Hankel transform was given by

hankel(f(t), t, s, nu) = Int(f(t)*sqrt(s*t)*BesselJ(nu, s*t), t = 0 .. infinity)

where BesselJ(nu, s*t) is the BesselJ(nu, s*t) function. This definition, sometimes called alternative definition of the Hankel transform, has the inconvenience of the square root sqrt(s*t) in the integrand, complicating the form of the hankel transform for the Laplacian in cylindrical coordinates. On the other hand, the definition more frequently used in the literature is

 hankel(f(t), t, s, nu) = Int(f(t)*t*BesselJ(nu, s*t), t = 0 .. infinity)

With it, the Hankel transform of diff(u(r, t), r, r)+(diff(u(r, t), r))/r+diff(u(r, t), t, t) is given by the simple ODE form d^2*`ℍ`(k, t)/dt^2-k^2*`ℍ`(k, t). Not just this transform but several other ones acquire a simpler form with the definition that does not have a square root in the integrand.

So the idea is to align Maple with this simpler definition, while keeping the previous definition as an alternative. Hence, by default, when you load the inttrans package, the new definition in use for the Hankel transform is

hankel(f(t), t, s, nu); % = convert(%, Int)

hankel(f(t), t, s, nu) = Int(f(t)*t*BesselJ(nu, s*t), t = 0 .. infinity)

(3.1)

You can change this default so that Maple works with the alternative definition as in previous releases.  For that purpose, use the new inttrans:-setup command (which you can also use to query about the definition in use at any moment):

setup(alternativehankeldefinition)

alternativehankeldefinition = false

(3.2)

This change in definition is automatically taken into account by other parts of the Maple library using the Hankel transform. For example, the differentiation rule with the new definition is

(%diff = diff)(hankel(f(t), t, z, nu), z)

%diff(hankel(f(t), t, z, nu), z) = -hankel(t*f(t), t, z, nu+1)+nu*hankel(f(t), t, z, nu)/z

(3.3)

This differentiation rule resembles (is connected to) the differentiation rule for BesselJ, and this is another advantage of the new definition.

(%diff = diff)(BesselJ(nu, z), z)

%diff(BesselJ(nu, z), z) = -BesselJ(nu+1, z)+nu*BesselJ(nu, z)/z

(3.4)

Furthermore, several transforms have acquired a simpler form, as for example:

`assuming`([(%hankel = hankel)(exp(I*a*r)/r, r, k, 0)], [a > 0, k < a])

%hankel(exp(I*a*r)/r, r, k, 0) = 1/(-a^2+k^2)^(1/2)

(3.5)

Let's compare: make the definition be as in previous releases.

setup(alternativehankeldefinition = true)

alternativehankeldefinition = true

(3.6)

hankel(f(t), t, s, nu); % = convert(%, Int)

hankel(f(t), t, s, nu) = Int(f(t)*(s*t)^(1/2)*BesselJ(nu, s*t), t = 0 .. infinity)

(3.7)

The differentiation rule with the previous (alternative) definition was not as simple:

(%diff = diff)(hankel(f(t), t, s, nu), s)

%diff(hankel(f(t), t, s, nu), s) = -hankel(t*f(t), t, s, nu+1)+nu*hankel(f(t), t, s, nu)/s+(1/2)*hankel(f(t), t, s, nu)/s

(3.8)

And the transform (3.5) was also not so simple:

`assuming`([(%hankel = hankel)(exp(I*a*r)/r, r, k, 0)], [a > 0, k < a])

%hankel(exp(I*a*r)/r, r, k, 0) = (I*a*hypergeom([3/4, 3/4], [3/2], a^2/k^2)*GAMMA(3/4)^4+Pi^2*k*hypergeom([1/4, 1/4], [1/2], a^2/k^2))/(k*Pi*GAMMA(3/4)^2)

(3.9)

Reset to the new default value of the definition.

setup(alternativehankeldefinition = false)

alternativehankeldefinition = false

(3.10)

hankel(f(t), t, s, nu); % = convert(%, Int)

hankel(f(t), t, s, nu) = Int(f(t)*t*BesselJ(nu, s*t), t = 0 .. infinity)

(3.11)

More integral transform results

 

 

The revision of the integral transforms includes also filling gaps: previous transforms that were not being computed are now computed. Still with the Hankel transform, consider the operators

`D/t` := proc (u) options operator, arrow; (diff(u, t))/t end proc
formula_plus := t^(-nu)*(`D/t`@@m)(t^(m+nu)*u(t))

formula_minus := t^nu*(`D/t`@@m)(t^(m-nu)*u(t))

 

Being able to transform these operators into algebraic expressions or differential equations of lower order is key for solving PDE problems with Boundary Conditions.

 

Consider, for instance, this ODE

setup(computederivatives = false)

computederivatives = false

(4.1)

simplify(eval(formula_minus, [nu = 6, m = 3]))

((diff(diff(diff(u(t), t), t), t))*t^3-12*(diff(diff(u(t), t), t))*t^2+57*(diff(u(t), t))*t-105*u(t))/t^3

(4.2)

Its Hankel transform is a simple algebraic expression

hankel(((diff(diff(diff(u(t), t), t), t))*t^3-12*(diff(diff(u(t), t), t))*t^2+57*(diff(u(t), t))*t-105*u(t))/t^3, t, s, 6)

-s^3*hankel(u(t), t, s, 3)

(4.3)

An example with formula_plus

simplify(eval(formula_plus, [nu = 7, m = 4]))

((diff(diff(diff(diff(u(t), t), t), t), t))*t^4+38*(diff(diff(diff(u(t), t), t), t))*t^3+477*(diff(diff(u(t), t), t))*t^2+2295*(diff(u(t), t))*t+3465*u(t))/t^4

(4.4)

hankel(((diff(diff(diff(diff(u(t), t), t), t), t))*t^4+38*(diff(diff(diff(u(t), t), t), t))*t^3+477*(diff(diff(u(t), t), t))*t^2+2295*(diff(u(t), t))*t+3465*u(t))/t^4, t, s, 7)

s^4*hankel(u(t), t, s, 11)

(4.5)

In the case of hankel , not just differential operators but also several new transforms are now computable

hankel(1, r, k, nu)

piecewise(nu = 0, Dirac(k)/k, nu/k^2)

(4.6)

hankel(r^m, r, k, nu)

piecewise(And(nu = 0, m = 0), Dirac(k)/k, 2^(m+1)*k^(-m-2)*GAMMA(1+(1/2)*m+(1/2)*nu)/GAMMA((1/2)*nu-(1/2)*m))

(4.7)

NULL

Mellin and Hankel transform solutions for Partial Differential Equations with Boundary Conditions

 


In previous Maple releases, the Fourier and Laplace transforms were used to compute exact solutions to PDE problems with boundary conditions. Now, Mellin and Hankel transforms are also used for that same purpose.

 

Example:

pde := x^2*(diff(u(x, y), x, x))+x*(diff(u(x, y), x))+diff(u(x, y), y, y) = 0

iv := u(x, 0) = 0, u(x, 1) = piecewise(0 <= x and x < 1, 1, 1 < x, 0)

sol := pdsolve([pde, iv])

u(x, y) = invmellin(sin(s*y)/(sin(s)*s), s, x)

(5.1)


As usual, you can let pdsolve choose the solving method, or indicate the method yourself:

pde := diff(u(r, t), r, r)+(diff(u(r, t), r))/r+diff(u(r, t), t, t) = -Q__0*q(r)
iv := u(r, 0) = 0

pdsolve([pde, iv])

u(r, t) = Q__0*(-hankel(exp(-s*t)*hankel(q(r), r, s, 0)/s^2, s, r, 0)+hankel(hankel(q(r), r, s, 0)/s^2, s, r, 0))

(5.2)

It is sometimes preferable to see these solutions in terms of more familiar integrals. For that purpose, use

convert(u(r, t) = Q__0*(-hankel(exp(-s*t)*hankel(q(r), r, s, 0)/s^2, s, r, 0)+hankel(hankel(q(r), r, s, 0)/s^2, s, r, 0)), Int, only = hankel)

u(r, t) = Q__0*(-(Int(exp(-s*t)*(Int(q(r)*r*BesselJ(0, r*s), r = 0 .. infinity))*BesselJ(0, r*s)/s, s = 0 .. infinity))+Int((Int(q(r)*r*BesselJ(0, r*s), r = 0 .. infinity))*BesselJ(0, r*s)/s, s = 0 .. infinity))

(5.3)

An example where the hankel transform is computable to the end:

pde := c^2*(diff(u(r, t), r, r)+(diff(u(r, t), r))/r) = diff(u(r, t), t, t)
iv := u(r, 0) = A*a/(a^2+r^2)^(1/2), (D[2](u))(r, 0) = 0
NULL

`assuming`([pdsolve([pde, iv], method = Hankel)], [r > 0, t > 0, a > 0])

u(r, t) = (1/2)*A*a*((-c^2*t^2+(2*I)*a*c*t+a^2+r^2)^(1/2)+(-c^2*t^2-(2*I)*a*c*t+a^2+r^2)^(1/2))/((-c^2*t^2-(2*I)*a*c*t+a^2+r^2)^(1/2)*(-c^2*t^2+(2*I)*a*c*t+a^2+r^2)^(1/2))

(5.4)

``


 

Download Integral_Transforms_(revamped).mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

When_A=1_and_M=0_1_2_4.mw

Can you help me please. i have a problem with the coding where i obtain the graph different with the journal that i have been referred to, Here is the codes, the graph that i obtained and the grpah from the journal. Thank for your help.
 

restart

kernelopts(version)

with(plots)

A := 1

M := 0

``

Eq1 := diff(f(eta), `$`(eta, 3))+(diff(f(eta), `$`(eta, 2)))*f(eta)-(diff(f(eta), eta))^2-M.(diff(f(eta), `$`(eta, 1)))-A.(diff(f(eta), `$`(eta, 1))+1/2.eta.(diff(f(eta), `$`(eta, 2)))) = 0

diff(diff(diff(f(eta), eta), eta), eta)+(diff(diff(f(eta), eta), eta))*f(eta)-(diff(f(eta), eta))^2-(diff(f(eta), eta))-(1/2)*(eta.(diff(diff(f(eta), eta), eta))) = 0

(1)

``

bcs := f(0) = 0, (D(f))(0) = 1, ((D@@2)(f))(0) = 0

dsys := {Eq1, bcs}

dsol := dsolve(dsys, numeric)

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 26, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..63, {(1) = 3, (2) = 3, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 1, (19) = 30000, (20) = 0, (21) = 1, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = 0.2523829377920773e-2, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..3, {(1) = .0, (2) = 1.0, (3) = .0}, datatype = float[8], order = C_order)), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..3, {(1) = .1, (2) = .1, (3) = .1}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, 1..3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0}, datatype = float[8], order = C_order), Array(1..3, 1..3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, 1..3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0}, datatype = float[8], order = C_order), Array(1..3, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = 0, (2) = 0, (3) = 0}, datatype = integer[8]), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0}, datatype = float[8], order = C_order)]), ( 8 ) = ([Array(1..3, {(1) = .0, (2) = 1.0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = 1.0, (2) = .0, (3) = 2.0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = f(eta), Y[2] = diff(f(eta),eta), Y[3] = diff(diff(f(eta),eta),eta)]`; YP[3] := -Y[3]*Y[1]+Y[2]^2+Y[2]+(1/2)*(X.Y[3]); YP[1] := Y[2]; YP[2] := Y[3]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = f(eta), Y[2] = diff(f(eta),eta), Y[3] = diff(diff(f(eta),eta),eta)]`; YP[3] := -Y[3]*Y[1]+Y[2]^2+Y[2]+(1/2)*(X.Y[3]); YP[1] := Y[2]; YP[2] := Y[3]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0)  ] ))  ] ); _y0 := Array(0..3, {(1) = 0., (2) = 0., (3) = 1.}); _vmap := array( 1 .. 3, [( 1 ) = (1), ( 2 ) = (2), ( 3 ) = (3)  ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) end if; `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 100 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch:  end try else try procname(procname("right")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch:  end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 100 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error  end try; if _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; if _src = 0 and 100 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see <a href='http://www.maplesoft.com/support/help/search.aspx?term=dsolve,maxfun' target='_new'>?dsolve,maxfun</a> for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see <a href='http://www.maplesoft.com/support/help/search.aspx?term=dsolve,maxfun' target='_new'>?dsolve,maxfun</a> for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [eta, f(eta), diff(f(eta), eta), diff(diff(f(eta), eta), eta)], (4) = []}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol := 1; _ndsol := _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error  end try end proc

(2)

odeplot(dsol, [eta, f(eta)], view = [0 .. 6, 0 .. 1], color = red, axes = box)

Warning, cannot evaluate the solution further right of 7.3544767, maxfun limit exceeded (see ?dsolve,maxfun for details)

 

Warning, cannot evaluate the solution further left of -7.3544767, maxfun limit exceeded (see ?dsolve,maxfun for details)

 

 

``


 

Download When_A=1_and_M=0_1_2_4.mw

 

I am trying to solve a coupled pair of PDEs (Burgers eqns) for u(x,t) and v(x,t).  However, my ICs and BCs are being rejected at the (0,0) corner of the domain. 

I have uploaded the code in the attached at this link.

PDE_rejects_IBCs.mw

Can anyone help please? 

Melvin

I have managed to install Maple2017 and open the installer using the Linux terminal and everything else but I can't find the file to run maple anywhere. I have been opening and doing everything with the linux terminal but I can't open maple without the filename as chromebook are fairly useless for downloading anything, does anyone know the name of the file or a way to make it run on chromebook? Thanks for any help offered in advance

How to read and write video file in maple?

i want to read one video file and then manipulate this to another file and then play two files to compare in maple

if maple 12 can not , which version can do?

I am trying to get jacobian computed for lorenz equations, not showing matrix, using Maple 2019 newest version. any guidance?

 

 

 

 

Dear all

I have a set composed by vectors C(K), S(K) i would like to show that this set of vectors is orthogonal

That is, we pick any two different vectors and I compute the usual inner product and I verify if the result is zero or no

many thinks for your help  

 

 

orthogonal.mw

 

 

 

 

 

DE:=f'''(x)+f''(x)(cos(alpha)x+sin(alpha)f(x))/2=0;

ics:=f(0)=0,f'(0)=1,f'(x)=0 for x tends to infinity;

How to solve this DE im RK methods?

HI

Hot on the heels of my previous question https://www.mapleprimes.com/questions/227935-Multiply-Contents-Of-Vector-To-Get-Expected-Value#answer262336

I have attempted to maximize growth while keeping variance below a fixed level (with not much success), but 

can any of you maximize the growth of a 3 stock portfolio while minimizing the variance.

Efficient_Frontier.mw

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