is it possible to find the operator g(tanhx) = tra...

g3 := tanh(x+1);
a:=eval(diff(g3,x\$n)/n!, x=0) assuming n>=0:
tanhx := sum(a*x^n, n=0..infinity):
tanhx2 := subs(x^n=subs(_C1=0, subs(t=n!, g2))*x^n, tanhx):
diff(tanhx2, x) - tanhx2;

would like to find a operator to make it equalt to itself , a new differential operator for new transcendental function tanhx2

ACA 2017 - Differential Algebra for an extended...

by:

I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra - 2017" . It was a very interesting event. This second presentation, about "Differential algebra with mathematical functions, symbolic powers and anticommutative variables", describes a project I started working in 1997 and that is at the root of Maple's dsolve and pdsolve performance with systems of equations. It is a unique approach. Not yet emulated in any other computer algebra system.

At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.

Differential algebra with mathematical functions,

symbolic powers and anticommutative variables

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

Abstract:
Computer algebra implementations of Differential Algebra typically require that the systems of equations to be tackled be rational in the independent and dependent variables and their partial derivatives, and of course that , everything is commutative.

It is possible, however, to extend this computational domain and apply Differential Algebra techniques to systems of equations that involve arbitrary compositions of mathematical functions (elementary or special), fractional and symbolic powers, as well as anticommutative variables and functions. This is the subject of this presentation, with examples of the implementation of these ideas in the Maple computer algebra system and its ODE and PDE solvers.

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 Differential polynomial forms for mathematical functions (basic)
 Differential polynomial forms for compositions of mathematical functions
 Generalization to many variables
 Arbitrary functions of algebraic expressions
 Examples of the use of this extension to include mathematical functions
 Differential Algebra with anticommutative variables

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

How do I calculate the wedge product of differenti...

Dear all,

I have the following question, this code:

restart:
with(DifferentialGeometry):
DGsetup([w1,w2],N):
eq1 := ExteriorDerivative(w1);
eq2 := ExteriorDerivative(w1) &wedge ExteriorDerivative(w2);
eq1 &wedge eq2;

Gives the error:
Error, (in DifferentialGeometry:-Tools:-DGzero)  given degree, 3, exceeds that of frame dimension, 2

Unfortunately, I am not so familiar with differential geometry but as far as I know dw1  (dw1  dw2) = 0 should be correct.

best
baustamm1

How do I solve a differential equation in Maple?...

How do I solve a differential equation in maple? Is there a specific identity that you can use?

Total differentiation using PDEtools package...

```with(DifferentialGeometry):with(JetCalculus):
DGsetup([x],[u],E,5);
vars≔x,u,u[1],u[1,1],u[1,1,1];
PDEtools[declare](Q(vars));
TotalDiff(Q(vars),x);
TotalDiff(u[1,1],x);```

Hi everyone,

Recently I came across the total differentiation command in the PDEtools. For its

documentation, I used the following link

http://www.maplesoft.com/support/help/Maple/view.aspx?path=DifferentialGeometry/JetCalculus/TotalDiff

Unfortunately, when I try to replicate this it did not work as expected. I am getting the total derivative of the expression to be zero. I do not understand where I am going wrong.

You can find my code above. I am also attaching the screen shot of my maple file.

I would really appreciate if someone could help me out. Thanks for your help.

Digitizing mathematics: ODEs, Special Functions...

by: Maple

The material below was presented in the "Semantic Representation of Mathematical Knowledge Workshop", February 3-5, 2016 at the Fields Institute, University of Toronto. It shows the approach I used for “digitizing mathematical knowledge" regarding Differential Equations, Special Functions and Solutions to Einstein's equations. While for these areas using databases of information helps (for example textbooks frequently contain these sort of databases), these are areas that, at the same time, are very suitable for using algorithmic mathematical approaches, that result in much richer mathematics than what can be hard-coded into a database. The material also focuses on an interesting cherry-picked collection of Maple functionality, that I think is beautiful, not well know, and seldom focused inter-related as here.

 Digitizing of special functions, differential equations, and solutions to Einstein’s equations within a computer algebra system   Edgardo S. Cheb-Terrab Physics, Differential Equations and Mathematical Functions, Maplesoft Editor, Computer Physics Communications

 • Big amounts of knowledge available to everybody in local machines or through the internet
 • Take advantage of basic computer functionality, like searching and editing

 • By digitizing mathematical knowledge inside appropriate computational contexts that understand about the topics, one can use the digitized knowledge to automatically generate more and higher level knowledge
 Challenges 1) how to identify, test and organize the key blocks of information,   2) how to access it: the interface,   3) how to mathematically process it to automatically obtain more information on demand
 Three examples

Mathematical Functions

"Mathematical functions, are defined by algebraic expressions. So consider algebraic expressions in general ..."

"Supporting information on definitions, identities, possible simplifications, integral forms, different types of series expansions, and mathematical properties in general"

 Examples
 General description
 References

Differential equation representation for generic nonlinear algebraic expressions - their use

"Compute differential polynomial forms for arbitrary systems of non-polynomial equations ..."

 The Differential Equations representing arbitrary algebraic expresssions
 Deriving knowledge: ODE solving methods
 Extending the mathematical language to include the inverse functions
 Solving non-polynomial algebraic equations by solving polynomial differential equations
 References

Branch Cuts of algebraic expressions

"Algebraically compute, and visualize, the branch cuts of arbitrary mathematical expressions"

 Examples
 References

Algebraic expresssions in terms of specified functions

"A conversion network for arbitrary mathematical expressions, to rewrite them in terms of different functions in flexible ways"

 Examples
 General description
 References

Symbolic differentiation of algebraic expressions

"Perform symbolic differentiation by combining different algebraic techniques, including functions of symbolic sequences and Faà di Bruno's formula"

 Examples
 References

Ordinary Differential Equations

"Beyond the concept of a database, classify an arbitrary ODE and suggest solution methods for it"

 General description
 Examples
 References

Exact Solutions to Einstein's equations

"The authors of "Exact solutions toEinstein's equations" reviewed more than 4,000 papers containing solutions to Einstein’s equations in the general relativity literature, organized the whole material into chapters according to the physical properties of these solutions. These solutions are key in the area of general relativity, are now all digitized and become alive in a worksheet"

The ability to search the database according to the physical properties of the solutions, their classification, or just by parts of keywords (old paradigm) changes the game.

More important, within a computer algebra system this knowledge becomes alive (new paradigm).

 • The solutions are turned active by a simple call to one commend, called the g_  spacetime metric.
 • Everything else gets automatically derived and set on the fly ( Christoffel symbols  , Ricci  and Riemann  tensors orthonormal and null tetrads , etc.)
 • Almost all of the mathematical operations one can perform on these solutions are implemented as commands in the Physics  and DifferentialGeometry  packages.
 • All the mathematics within the Maple library are instantly ready to work with these solutions and derived mathematical objects.

Finally, in the Maple PDEtools package , we have all the mathematical tools to tackle the equivalence problem around these solutions.

 Examples
 References

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

scheme of fractional differential equation in mapl...

i want a scheme of fractional differential equation so that i solve my questions and make a code of it.

word needed for definition...

what do i call a homogenous  differential equation that is the linear sum of "N" differential of unique classification? ie, the implicit construction of a third homogenous differential by the summation of two known, is it the span of the solution sets of the first two or union? i prefer span because well that leaves the door open for multivariate differential basis definitions, non commutative groups like sets of square matrices and all of the other extra arousing subject content.

Issues with differential and D operator...

Hi,

I am trying to do a numerical approximate method 'method of multiple scales'. And in doing so I am getting following equation.

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Now I need to substitute x_1(T0,T1,T2)=R(T1,T2)sin(omega*T0+phi(T1,T2)) in the expression and evaluate it.  But on substituting, it is not solving for the 'D' operator. In a similar line if I will have a differential term like

D1(x1)(T0-tau_1,T1,T2) and I have to substitute x1(T0-tau_1,T1,T2) then how can i do it?

Thanks and regards

Sunit

thanks

 (1)

How to deal with operator with both differential a...

In my study, I often need to verify that two operator is symmetric i.e. [P,Q]=PQ-QP=0, where A and D are operator polynomial such like  D2+4u+uxD-1 multiply with D3+uD+ux,where D is differential operator.

I tried to use the Ore_package which can easily deal with the operator polynomial without integral(i.e. D-1 term), so in my case , how to deal with operator with both differential and integral?

Initial value problem - dsolve does not give parti...

Trying to solve this IVP of the SHO  (second order linear costant-coefficient).

Everything works fine until I come to the solving even after using dsolve with initial conditions (even using the differential operator D in the initial conditions)  , the answer still contains _C1, an unknown constant.

The full worksheet is below.  The code for dsolve is:

sol3 := dsolve(subs(par1, {de1, D(x)*0 = 0, x(0) = 1}), x(t));

Hoping you can help with a solution.

PDE solutions: when are they "general"?

by: Maple

Hi,
The latest update to the differential equations Maple libraries (this week, can be downloaded from the Maplesoft R&D webpage for Differential Equations and Mathematical functions) includes new functionality in pdsolve, regarding whether the solution for a PDE or PDE system is or not a general solution.

In brief, a general solution of a PDE in 1 unknown, that has differential order N, and where the unknown depends on M independent variables, involves N arbitrary functions of M-1 arguments. It is not entirely evident how to extend this definition in the case of a coupled, possibly nonlinear PDE system. However, using differential algebra techniques (automatically used by pdsolve when tackling a PDE system), that extension to define a general solution for a DE system is possible, and also when the system involves ODEs and PDEs, and/or algebraic (that is, non-differential) equations, and/or inequations of the form  involving the unknowns, and all of this in the presence of mathematical functions (based on the use of Maple's PDEtools:-dpolyform). This is a very nice case were many different advanced developments come together to naturally solve a problem that otherwise would be rather difficult.

The issues at the center of this Maple development/post are then:

a) How do you know whether a PDE or PDE system solution returned is a general solution?

b) How could you indicate to pdsolve that you are only interested in a general PDE or PDE system solution?

The answer to a) is now always present in the last line of the userinfo. So input infolevel[pdsolve] := 3 before calling pdsolve, and check what the last line of the userinfo displayed tells.

The answer to b) is a new option, generalsolution, implemented in pdsolve so that it either returns a general solution or otherwise it returns NULL. If you do not use this new option, then pdsolve works as always: first it tries to compute a general solution and if it fails in doing that it tries to compute a particular solution by separating the variables in different ways, or computing a traveling wave solution or etc. (a number of other well known methods).

The examples that follow are from the help page pdsolve,system, and show both the new userinfo telling whether the solution returned is a general one and the option generalsolution at work.The examples are all of differential equation systems but the same userinfos and generalsolution option work as well in the case of a single PDE.

Example 1.

Solve the determining PDE system for the infinitesimals of the symmetry generator of example 11 from Kamke's book . Tell whether the solution computed is or not a general solution.

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The PDE system satisfied by the symmetries of Kamke's ODE example number 11 is

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This is a second order linear PDE system, with two unknowns  and four equations. Its general solution is given by the following, where we now can tell that the solution is a general one by reading the last line of the userinfo. Note that because the system is overdetermined, a general solution in this case does not involve any arbitrary function

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 -> Solving ordering for the dependent variables of the PDE system: [xi(x,y), eta(x,y)] -> Solving ordering for the independent variables (can be changed using the ivars option): [x, y] tackling triangularized subsystem with respect to xi(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to eta(x,y) <- Returning a *general* solution
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Next we indicate to pdsolve that  and  are parameters of the problem, and that we want a solution for , making more difficult to identify by eye whether the solution returned is or not a general one. Again the last line of the userinfo tells that pdsolve's solution is indeed a general one

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 -> Solving ordering for the dependent variables of the PDE system: [r, n, xi(x,y), eta(x,y)] -> Solving ordering for the independent variables (can be changed using the ivars option): [x, y] tackling triangularized subsystem with respect to r tackling triangularized subsystem with respect to n tackling triangularized subsystem with respect to xi(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to eta(x,y) tackling triangularized subsystem with respect to r tackling triangularized subsystem with respect to n tackling triangularized subsystem with respect to xi(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to eta(x,y) tackling triangularized subsystem with respect to r tackling triangularized subsystem with respect to n tackling triangularized subsystem with respect to xi(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to eta(x,y) tackling triangularized subsystem with respect to n tackling triangularized subsystem with respect to xi(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to eta(x,y) tackling triangularized subsystem with respect to n tackling triangularized subsystem with respect to xi(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to eta(x,y) tackling triangularized subsystem with respect to xi(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to eta(x,y) <- Returning a *general* solution
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Example 2.

Compute the solution of the following (linear) overdetermined system involving two PDEs, three unknown functions, one of which depends on 2 variables and the other two depend on only 1 variable.

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The solution for the unknowns G, H, is given by the following expression, were again determining whether this solution, that depends on 3 arbitrary functions, , is or not a general solution, is non-obvious.

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 -> Solving ordering for the dependent variables of the PDE system: [F(r,s), H(r), G(s)] -> Solving ordering for the independent variables (can be changed using the ivars option): [r, s] tackling triangularized subsystem with respect to F(r,s) First set of solution methods (general or quasi general solution) Trying differential factorization for linear PDEs ... differential factorization successful. First set of solution methods successful tackling triangularized subsystem with respect to H(r) tackling triangularized subsystem with respect to G(s) <- Returning a *general* solution
 (1.6)
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Example 3.

Compute the solution of the following nonlinear system, consisting of Burger's equation and a possible potential.

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We see that in this case the solution returned is not a general solution but two particular ones; again the information is in the last line of the userinfo displayed

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 -> Solving ordering for the dependent variables of the PDE system: [v(x,t), u(x,t)] -> Solving ordering for the independent variables (can be changed using the ivars option): [x, t] tackling triangularized subsystem with respect to v(x,t) tackling triangularized subsystem with respect to u(x,t) First set of solution methods (general or quasi general solution) Second set of solution methods (complete solutions) Trying methods for second order PDEs Third set of solution methods (simple HINTs for separating variables) PDE linear in highest derivatives - trying a separation of variables by * HINT = * Fourth set of solution methods Trying methods for second order linear PDEs Preparing a solution HINT ... Trying HINT = _F1(x)*_F2(t) Fourth set of solution methods Preparing a solution HINT ... Trying HINT = _F1(x)+_F2(t) Trying travelling wave solutions as power series in tanh ... * Using tau = tanh(t*C[2]+x*C[1]+C[0]) * Equivalent ODE system: {C[1]^2*(tau^2-1)^2*diff(diff(u(tau),tau),tau)+(2*C[1]^2*(tau^2-1)*tau+2*u(tau)*C[1]*(tau^2-1)+C[2]*(tau^2-1))*diff(u(tau),tau)} * Ordering for functions: [u(tau)] * Cases for the upper bounds: [[n[1] = 1]] * Power series solution [1]: {u(tau) = tau*A[1,1]+A[1,0]} * Solution [1] for {A[i, j], C[k]}: [[A[1,1] = 0], [A[1,0] = -1/2*C[2]/C[1], A[1,1] = -C[1]]] travelling wave solutions successful. tackling triangularized subsystem with respect to v(x,t) First set of solution methods (general or quasi general solution) Trying differential factorization for linear PDEs ... Trying methods for PDEs "missing the dependent variable" ... Second set of solution methods (complete solutions) Trying methods for second order PDEs Third set of solution methods (simple HINTs for separating variables) PDE linear in highest derivatives - trying a separation of variables by * HINT = * Fourth set of solution methods Trying methods for second order linear PDEs Preparing a solution HINT ... Trying HINT = _F1(x)*_F2(t) Third set of solution methods successful tackling triangularized subsystem with respect to u(x,t) <- Returning a solution that *is not the most general one*
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This example is also good for illustrating the other related new feature: one can now request to pdsolve to only compute a general solution (it will return NULL if it cannot achieve that). Turn OFF userinfos and try with this example

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This returns NULL:

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Example 4.

Another where the solution returned is particular, this time for a linear system, conformed by 38 PDEs, also from differential equation symmetry analysis

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There are 38 coupled equations

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When requesting a general solution pdsolve returns NULL:

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A solution that is not a general one, is however computed by default if calling pdsolve without the generalsolution option. In this case again the last line of the userinfo tells that the solution returned is not a general solution

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 (1.11)
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 -> Solving ordering for the dependent variables of the PDE system: [eta[1](x,y,z,t,u), xi[1](x,y,z,t,u), xi[2](x,y,z,t,u), xi[3](x,y,z,t,u), xi[4](x,y,z,t,u)] -> Solving ordering for the independent variables (can be changed using the ivars option): [t, x, y, z, u] tackling triangularized subsystem with respect to eta[1](x,y,z,t,u) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F1(x,y,z,t), _F2(x,y,z,t)] -> Solving ordering for the independent variables (can be changed using the ivars option): [t, x, y, z, u] tackling triangularized subsystem with respect to _F1(x,y,z,t) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F3(x,y,z), _F4(x,y,z)] -> Solving ordering for the independent variables (can be changed using the ivars option): [x, y, z, t] tackling triangularized subsystem with respect to _F3(x,y,z) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to _F4(x,y,z) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F5(y,z), _F6(y,z)] -> Solving ordering for the independent variables (can be changed using the ivars option): [y, z, x] tackling triangularized subsystem with respect to _F5(y,z) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to _F6(y,z) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F7(z), _F8(z)] -> Solving ordering for the independent variables (can be changed using the ivars option): [z, y] tackling triangularized subsystem with respect to _F7(z) tackling triangularized subsystem with respect to _F8(z) tackling triangularized subsystem with respect to _F2(x,y,z,t) First set of solution methods (general or quasi general solution) Trying differential factorization for linear PDEs ... Trying methods for PDEs "missing the dependent variable" ... Second set of solution methods (complete solutions) Third set of solution methods (simple HINTs for separating variables) PDE linear in highest derivatives - trying a separation of variables by * HINT = * Fourth set of solution methods Preparing a solution HINT ... Trying HINT = _F3(x)*_F4(y)*_F5(z)*_F6(t) Third set of solution methods successful tackling triangularized subsystem with respect to xi[1](x,y,z,t,u) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F1(x,z,t), _F2(x,z,t)] -> Solving ordering for the independent variables (can be changed using the ivars option): [t, x, z, y] tackling triangularized subsystem with respect to _F1(x,z,t) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to _F2(x,z,t) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F3(x,t), _F4(x,t)] -> Solving ordering for the independent variables (can be changed using the ivars option): [t, x, z] tackling triangularized subsystem with respect to _F3(x,t) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to _F4(x,t) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F5(x), _F6(x)] -> Solving ordering for the independent variables (can be changed using the ivars option): [x, t] tackling triangularized subsystem with respect to _F5(x) tackling triangularized subsystem with respect to _F6(x) tackling triangularized subsystem with respect to xi[2](x,y,z,t,u) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F1(t), _F2(t)] -> Solving ordering for the independent variables (can be changed using the ivars option): [t, z] tackling triangularized subsystem with respect to _F1(t) tackling triangularized subsystem with respect to _F2(t) tackling triangularized subsystem with respect to xi[3](x,y,z,t,u) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to xi[4](x,y,z,t,u) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful <- Returning a solution that *is not the most general one*
 (1.12)
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 (1.13)

Example 5.

Finally, the new userinfos also tell whether a solution is or not a general solution when working with PDEs that involve anticommutative variables  set using the Physics  package

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 (1.14)

Set first  and  as suffixes for variables of type/anticommutative  (see Setup )

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 (1.15)

A PDE system example with two unknown anticommutative functions of four variables, two commutative and two anticommutative; to avoid redundant typing in the input that follows and redundant display of information on the screen let's use PDEtools:-diff_table   PDEtools:-declare

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 (1.16)
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 (1.17)

Consider the system formed by these two PDEs (because of the q diff_table just defined, we can enter derivatives directly using the function's name indexed by the differentiation variables)

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 (1.18)
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 (1.19)

The solution returned for this system is indeed a general solution

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 -> Solving ordering for the dependent variables of the PDE system: [_F4(x,y), _F2(x,y), _F3(x,y)] -> Solving ordering for the independent variables (can be changed using the ivars option): [x, y] tackling triangularized subsystem with respect to _F4(x,y) tackling triangularized subsystem with respect to _F2(x,y) tackling triangularized subsystem with respect to _F3(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. HINT = _F6(x)+_F5(y) Trying HINT = _F6(x)+_F5(y) HINT is successful First set of solution methods successful <- Returning a *general* solution
 (1.20)
 >

This solution involves an anticommutative constant , analogous to the commutative constants  where n is an integer.

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

How can I apply this differential in Maple?...

hi

how i can apply this differential in maple?

thabks...

Fractional differential equation...

Hi!

I am simulate the code for fractional differential equation. But the out put is not wright...
sir_(2).mw

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