mmcdara

A way to get eq 11...

Answered: mmcdara 7891

February 16 2025

3 7

The soution is based upon two procedures f and `w'/w` :

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(1)

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(2)

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(3)

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(4)

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# Procedure 1

f := proc(m, n)
  local amn := (m, n) -> cat('a__', nprintf("%a,%a", (m, n)));
  if m=0 and n=0 then
    amn(m, n);
  elif m=0 then
    amn(m, n)*diff(log(w(x, t)), t$n);
  elif n=0 then
    amn(m, n)*diff(log(w(x, t)), x$m);
  else
    amn(m, n)*diff(log(w(x, t)), [x$m, t$n]);
  end if:
end proc:

>	eq_8 := u(x, t) = add(f(n, 0), n=0..2); A := convert( remove(type, indets(eq_8), function) minus indets(pde, name), list);

u(x, t) = `a__0,0`+`a__1,0`*(diff(w(x, t), x))/w(x, t)+`a__2,0`*((diff(diff(w(x, t), x), x))/w(x, t)-(diff(w(x, t), x))^2/w(x, t)^2)

(5)

>	eq_8w := collect(eval(pde, eq_8), A)

(6)

>	# Procedure 2 `w'/w` := d -> eval( add(select(has, [op(lhs(expand(eq_8w)))], diff(w(x, t), x)^d)) , w(x, t) = exp(x) ):

>	eq_10 := {`w'/w`(5) = 0, `w'/w`(4) = 0}

${-2*`a__2,0`^2+24*`a__2,0`*delta = 0, 3*`a__1,0`*`a__2,0`-6*`a__1,0`*delta = 0}$

(7)

>	sol := solve(eq_10, indets(eq_10) minus {delta}); sol := remove(i -> rhs~(i) = {0}, {sol})[]

${`a__1,0` = 0, `a__2,0` = 0}, {`a__1,0` = 0, `a__2,0` = 12*delta}$

${`a__1,0` = 0, `a__2,0` = 12*delta}$

(8)

>	eq_11 := eval(eq_8, sol);

u(x, t) = `a__0,0`+12*delta*((diff(diff(w(x, t), x), x))/w(x, t)-(diff(w(x, t), x))^2/w(x, t)^2)

(9)

>	# Verify this expression is equal to eq (11) from the paper: `a__0,0`+12deltadiff(log(w(x, t)), x$2)

`a__0,0`+12*delta*((diff(diff(w(x, t), x), x))/w(x, t)-(diff(w(x, t), x))^2/w(x, t)^2)

(10)

>

Download series-finding_mmcdara.mw

A solution...

Answered: mmcdara 7891

February 14 2025

2 1

>

>	Jac := [ seq(j = freeze(j), j in indets(Eq, function)) ]: EqFreezed := subs(Jac[], lhs(Eq)):

>

Dx := $(ldegree..degree)(EqFreezed, x);
Dy := $(ldegree..degree)(EqFreezed, y);
Dz := $(ldegree..degree)(EqFreezed, z);
Dt := $(ldegree..degree)(EqFreezed, t);

TC := table():
for dt in Dt do
  for dz in Dz do
    for dy in Dy do
      for dx in Dx do
        TC[x^dx*y^dy*z^dz*t^dt]
        :=
        simplify(add(select(i -> `and`(has~(i, [x^dx, y^dy, z^dz, t^dt])[]), [op(expand(EqFreezed))]))):
      end do:
    end do:
  end do:
end do:

(1)

>	# Coefficient of xyzt TC[xyzt]

(2)

>	# All 0 coefficients ZC, NZC := selectremove(i -> is(TC[i]=0), [indices(TC, nolist)]): print~(ZC):

(3)

>	# All non 0 coefficients print~(NZC):

(4)

>	# Example TC[y^2zt]; print(): thaw(%);

(4/25)*beta*y^2*`freeze/R0`*`freeze/R2`^2*w[1, u]*w[3, u]*w[t, 3]*w[y, 1]^2*w[z, 3]-beta*y^2*`freeze/R0`*`freeze/R1`^2*w[1, u]*w[3, u]*w[t, 3]*w[y, 1]^2*w[z, 3]-2*beta*y^2*w[1, u]^2*w[t, 1]*w[y, 1]^2*w[z, 1]

(4/25)*beta*y^2*JacobiCN(t*w[t, 3]+x*w[x, 3]+y*w[y, 3]+z*w[z, 3], 2/5)*JacobiSN(t*w[t, 3]+x*w[x, 3]+y*w[y, 3]+z*w[z, 3], 2/5)^2*w[1, u]*w[3, u]*w[t, 3]*w[y, 1]^2*w[z, 3]-beta*y^2*JacobiCN(t*w[t, 3]+x*w[x, 3]+y*w[y, 3]+z*w[z, 3], 2/5)*JacobiDN(t*w[t, 3]+x*w[x, 3]+y*w[y, 3]+z*w[z, 3], 2/5)^2*w[1, u]*w[3, u]*w[t, 3]*w[y, 1]^2*w[z, 3]-2*beta*y^2*w[1, u]^2*w[t, 1]*w[y, 1]^2*w[z, 1]

(5)

>

# This could also be useful.
#
# See LargeExpressions and Veilhelp pages and examples

with(LargeExpressions):

# All partial expressions have the same form

compact_x := collect(EqFreezed, x, Veil[Cx] ));
compact_y := collect(EqFreezed, y, Veil[Cy] );
compact_z := collect(EqFreezed, z, Veil[Cz] );
compact_t := collect(EqFreezed, t, Veil[Ct] );

(6)

Download collection_of_coefficients_mmcdara.mw

ODE is not an ordinary differential equa...

Answered: mmcdara 7891

February 13 2025

1 6

An ordinary differential equation of order n writes

diff(phi(x), x$n) = F(x, diff(phi(x), x), ..., diff(phi(x), x$(n-1))

Your differential equation (DE) is

(diff(phi(x), x)^2 = -v(phi(x))

and is not an ODE.

Undortunately dsolve/numeric can handle only ODEs, not DEs.

One idea could be to rewrite your DE this way where siga is either -1 or +1 dependeing on the branch you want to select.

diff(phi(x), x = sigma * sqrt(-v(phi(x)))

Nevertheless this will not give you a correct result as this simple example proves:

( initial worksheet where @dharr detected a typo Download DE_not_ODE.mw)

Edited worksheet DE_not_ODE_edited.mw

Continuation, maxmesh and absolute error...

Answered: mmcdara 7891

February 13 2025

3 2

Every time you get an error while solving numerically a BVP yiu must look the dsolve[numeric][BVP] help page.
The Error, (in dsolve/numeric/bvp) initial Newton iteration is not converging is a classic for which this help page proposes a few cures (search the word Newton).
Another important help page is dsolve[numeric_bvp,advanced].

Here is the solution to yoiur problem (I just treated b1 and b2: end the job by adjusting the continuation term, maxmesh and abserr as you want.

VC1_mmcdara.mw

>

PDEtools:-declare(f(x));

(1)

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J[1] := mu[f]/(1-phi)^2.5:

>

eq1 := J[1]*(diff(f(x), x, x, x))/J[2]+(7/4)*f(x)*(diff(f(x), x, x))-(1/2)*(diff(f(x), x))^2+J[3]*theta/J[2]-M^2*(diff(f(x), x))/((m^2+1)*J[2])-J[1]*K[p]*(diff(f(x), x))/J[2] = 0;

(diff(diff(diff(f(x), x), x), x))/((1-phi)^2.5*(9997.1-1064.100000*phi))+(7/4)*f(x)*(diff(diff(f(x), x), x))-(1/2)*(diff(f(x), x))^2+(209939.10-195020.9900*phi)*theta/(9997.1-1064.100000*phi)-M^2*(diff(f(x), x))/((m^2+1)*(9997.1-1064.100000*phi))-K[p]*(diff(f(x), x))/((1-phi)^2.5*(9997.1-1064.100000*phi)) = 0

(2)

>

eq2 := (N[R]+J[4])*(diff(theta(x), x, x))/(Pr*J[5])+(7/4)*f(x)*(diff(theta(x), x))+M^2*Ec*(diff(f(x), x))^2/((m^2+1)*J[5])+N[b]*(diff(theta(x), x))*(diff(phi(x), x))/J[5]+N[t]*(diff(theta(x), x))^2/J[5] = 0;

(N[R]+.6130*(402.2260+800.7740*phi)/(402.2260-400.3870*phi))*(diff(diff(theta(x), x), x))/(Pr*(41777880.9-38338675.90*phi))+(7/4)*f(x)*(diff(theta(x), x))+M^2*Ec*(diff(f(x), x))^2/((m^2+1)*(41777880.9-38338675.90*phi))+N[b]*(diff(theta(x), x))*(diff(phi(x), x))/(41777880.9-38338675.90*phi)+N[t]*(diff(theta(x), x))^2/(41777880.9-38338675.90*phi) = 0

(3)

>

eq3 := diff(phi(x), x, x)+(7/4)*Sc*f*(diff(phi(x), x))+N[t]*(diff(theta(x), x, x))/N[b]-Sc*lambda[c]*phi = 0;

diff(diff(phi(x), x), x)+(7/4)*Sc*f*(diff(phi(x), x))+N[t]*(diff(diff(theta(x), x), x))/N[b]-Sc*lambda[c]*phi = 0

(4)

>

(5)

>

a1 := [phi = 0, M = .5, m = .3, lambda[c] = .3, Pr = 1.5, K[p] = .2, N[R] = 1.5, Ec = .3, N[t] = .2, N[b] = .2, Sc = 1.2];

(6)

>

(7)

>	$indets(b1, function); b1_bcs := select(has, {bcs}, f); c1 := dsolve(`union`(eval({b1}, theta = 1), b1_bcs), numeric)$

Error, (in dsolve/numeric/bvp) initial Newton iteration is not converging

>

# Inspired from dsolve[numeric_bvp,advanced] equations (8) to (11)

eval({b1}, theta = 1);

new_ode := {
(10*(1-lambda) + 0.0001000290084) * (diff(f(x), x, x, x))
+(7/4)*f(x)*(diff(f(x), x, x))-(1/2)*(diff(f(x), x))^2+21.00000000-0.4294823481e-4*(diff(f(x), x)) = 0
};

c1 := dsolve(new_ode union b1_bcs, numeric, continuation=lambda, maxmesh=2^13, abserr=1e-4);

${0.1000290084e-3*(diff(diff(diff(f(x), x), x), x))+(7/4)*f(x)*(diff(diff(f(x), x), x))-(1/2)*(diff(f(x), x))^2+21.00000000-0.4294823481e-4*(diff(f(x), x)) = 0}$

${(10.00010003-10*lambda)*(diff(diff(diff(f(x), x), x), x))+(7/4)*f(x)*(diff(diff(f(x), x), x))-(1/2)*(diff(f(x), x))^2+21.00000000-0.4294823481e-4*(diff(f(x), x)) = 0}$

(8)

>	p1 := odeplot(c1, [x, diff(f(x), x)], 0 .. 8, colour = blue, legend="c1"):

>

b2_bcs := select(has, {bcs}, f);

eval({b2}, theta = 1);

new_ode := {
(10*(1-lambda) + 0.0001143232223) * (diff(f(x), x, x, x))
+(7/4)*f(x)*(diff(f(x), x, x))-(1/2)*(diff(f(x), x))^2+20.13175426-0.4592983151e-4*(diff(f(x), x)) = 0
};

c2 := dsolve(new_ode union b2_bcs, numeric, continuation=lambda, maxmesh=2^13, abserr=1e-4);

${0.1143232223e-3*(diff(diff(diff(f(x), x), x), x))+(7/4)*f(x)*(diff(diff(f(x), x), x))-(1/2)*(diff(f(x), x))^2+20.13175426-0.4592983151e-4*(diff(f(x), x)) = 0}$

${(10.00011432-10*lambda)*(diff(diff(diff(f(x), x), x), x))+(7/4)*f(x)*(diff(diff(f(x), x), x))-(1/2)*(diff(f(x), x))^2+20.13175426-0.4592983151e-4*(diff(f(x), x)) = 0}$

(9)

>	p2 := odeplot(c2, [x, diff(f(x), x)], 0 .. 8, colour = green, legend="c2"):

>	display(p1, p2)

>

# A method to get the coefficient of diff(f(x), x$3) and apply it the continuation term.
#
# Example

ini_ode := lhs(eval(b1, theta = 1));
s, r := selectremove(has, [op(ini_ode)], diff(f(x), x$3));

c3 := eval(s, diff(f(x), x$3)=1)[];

new_ode := {eval(s[], c3 = (10*(1-lambda) + c3)) + add(r) = 0};

c1 := dsolve(new_ode union b1_bcs, numeric, continuation=lambda, maxmesh=2^13, abserr=1e-4);

${(diff(diff(diff(f(x), x), x), x))*(10.00010003-10*lambda)+21.00000000+(7/4)*f(x)*(diff(diff(f(x), x), x))-(1/2)*(diff(f(x), x))^2-0.4294823481e-4*(diff(f(x), x)) = 0}$

(10)

>	# Proceed this way for all your equations. # # You can "play" with maxmesh (increase its value) and abserr (decrease its value)

${(diff(diff(diff(f(x), x), x), x))*(10.00010003-10*lambda)+21.00000000+(7/4)*f(x)*(diff(diff(f(x), x), x))-(1/2)*(diff(f(x), x))^2-0.4294823481e-4*(diff(f(x), x)) = 0}$

(11)

>

Download VC1_mmcdara.mw

Be careful...

Answered: mmcdara 7891

February 13 2025

1 2

Do not confuse the PDF of a random variable with the restriction of this PDF to the support of this same random variable.

Imposing assumptions is a simple trick to get the expression of the restriction of the PDF of r*x to the support of r*x, but this does not give you the expression of the PDF.

You can get the exact PDF by doing the things rigourously while avoiding any assumption (even if it is more lenghty than the previous trick, it is instructing, at least I think so):

restart;
with(IntegrationTools):
with(Statistics):
u := RandomVariable(Uniform(0,1)):
v := RandomVariable(Uniform(0, 2*Pi)):
r := sqrt(u):
x := cos(v):
w := r*x:

f__r := PDF(r, t):
f__x := PDF(x, t):
f__w := PDF(w, t):

J  := Int(op(f__w)):
d  := discont(f__x, t):
JS := Split(J, convert(d, list)):
K1 := op(2, JS):
K2 := Split(K1, 0);

f__w := simplify(map(value, K2));

Full details here rv_pdf_mmcdara.mw

Here is a trick...

Answered: mmcdara 7891

February 10 2025

1 0

>

restart

>	kernelopts(version)

(1)

>	e := xyz: c := x*z = 0;

(2)

>	# trick: define A = x*z E := eval(e, x=A/z); C := A = 0; `Could it be that e > 0 given c? ` = coulditbe(E > 0) assuming C, y::real; `Could it be that e < 0 given c? ` = coulditbe(E < 0) assuming C, y::real; `Could it be that e = 0 given c? ` = coulditbe(E = 0) assuming C, y::real;

(3)

Generalization

>	e := uvwxyz; c := uw*y = 0;

(4)

>

rule := isolate(lhs(c)=A, op(1, lhs(c))):

E := eval(e, rule):
C := eval(C, rule):

R := indets(e, name) minus indets(c, name):
T := seq(r::real, r in R):

`Could it be that e > 0 given c? ` = coulditbe(E > 0) assuming C, T;
`Could it be that e < 0 given c? ` = coulditbe(E < 0) assuming C, T;
`Could it be that e = 0 given c? ` = coulditbe(E = 0) assuming C, T;

(5)

Download A_Trick.mw

I propose you this simplification...

Answered: mmcdara 7891

February 10 2025

3 2

(equation (13) )

Note 1: an important point is the correct writting of p:

 p := {alpha = t^2, beta = exp(-t)};

instead of

p := {alpha(t) = t^2, beta = exp(-t)}

Note 2: I also use this"trick" (at least it enables having readable formulas)

alias(Z = Z(t)):
aux := Z = int((alpha + b*beta),t);

# and replacing all occurrences of int((alpha + b*beta),t) by Z likely helps 
# Maple in simplifying the successive relations

Note 3: for readibility I replace Qi by Q_i for each Q in {phi, psi, lambda} and i in {1, 2}

>

restart;

>	kernelopts(version);

(1)

>

>	interface(showassumed=0):

>	assume(x::real);assume(t::real);assume(lambda__1::complex);assume(b::real);

>	alias( psi__1 = psi__1(x,t), psi__2 = psi__2(x,t), phi__1 = phi__1(x,t), phi__2 = phi__2(x,t), beta = beta(t), alpha =alpha(t), Z = Z(t) );

(2)

>

aux := Z = int((alpha + b*beta),t);

rel := {psi__1 = exp((-I*lambda__1)*x - (1/(4*I*lambda__1))*Z), psi__2 = exp((I*lambda__1)*x + (1/(4*I*lambda__1))*Z), phi__1= exp((-I*conjugate(lambda__1))*x - (1/(4*I*conjugate(lambda__1)))*Z), phi__2 = exp((I*conjugate(lambda__1))*x + (1/(4*I*conjugate(lambda__1)))*Z)}

${phi__1 = exp(-I*conjugate(lambda__1)*x+((1/4)*I)*Z/conjugate(lambda__1)), phi__2 = exp(I*conjugate(lambda__1)*x-((1/4)*I)*Z/conjugate(lambda__1)), psi__1 = exp(-I*lambda__1*x+((1/4)*I)*Z/lambda__1), psi__2 = exp(I*lambda__1*x-((1/4)*I)*Z/lambda__1)}$

(3)

>	Bnum := psi__2phi__1conjugate(lambda__1) + psi__1lambda__1phi__2;

(4)

>	Bnumexp := map(normal, subs(rel,Bnum));

exp(-((1/4)*I)*(-4*lambda__1^2*x+Z)/lambda__1)*exp(((1/4)*I)*(-4*conjugate(lambda__1)^2*x+Z)/conjugate(lambda__1))*conjugate(lambda__1)+exp(((1/4)*I)*(-4*lambda__1^2*x+Z)/lambda__1)*lambda__1*exp(-((1/4)*I)*(-4*conjugate(lambda__1)^2*x+Z)/conjugate(lambda__1))

(5)

>	Den := -phi__1psi__2 - phi__2psi__1;

(6)

>	expDen := map(normal, subs(rel, Den))

-exp(((1/4)*I)*(-4*conjugate(lambda__1)^2*x+Z)/conjugate(lambda__1))*exp(-((1/4)*I)*(-4*lambda__1^2*x+Z)/lambda__1)-exp(-((1/4)*I)*(-4*conjugate(lambda__1)^2*x+Z)/conjugate(lambda__1))*exp(((1/4)*I)*(-4*lambda__1^2*x+Z)/lambda__1)

(7)

>	sr := Bnumexp/expDen; ratiosr := simplify(diff(sr, t), size);

(8)

>	ratiosr := eval(ratiosr, aux);

(9)

>	B := b - (4I/beta)ratiosr

(10)

>	p := {alpha = t^2, beta = exp(-t)};

${alpha = t^2, beta = exp(-t)}$

(11)

>	B1 := eval(B, p)

(12)

>	B1_1 := map(combine, B1, exp): NU := numer(op(2, B1_1)): DE := denom(op(2, B1_1)): DE := convert(simplify(DE, size), trig): B1_2 := op(1, B1) + NU/DE

b+(1/2)*(b*exp(-t)+t^2)*(-lambda__1+conjugate(lambda__1))^2*exp(t)/(cos((1/4)*(lambda__1-conjugate(lambda__1))*(-(1/3)*t^3-4*x*conjugate(lambda__1)*lambda__1+b/exp(t))/(lambda__1*conjugate(lambda__1)))^2*lambda__1*conjugate(lambda__1))

(13)

>

Download simplify_mmcdara_1.mw

It's not a bug...

Answered: mmcdara 7891

February 10 2025

1 0

x::real is too loose an assumption:

>

restart

>	plot( [cos(x), sqrt(1-sin(x)^2), -sqrt(1-sin(x)^2)] , x = 0..2*Pi , color=[red, green, blue] , style=[point, line, line] , legend=[typeset(cos(x)), typeset(sqrt(1-sin(x)^2)), typeset(-sqrt(1-sin(x)^2))] )

>	is(cos(x) = sqrt(1-sin(x)^2)) assuming Or(And(x >= 0, x <= Pi/2), And(x >= 3Pi/2, x <= 2Pi))

(1)

>	is(cos(x) = -sqrt(1-sin(x)^2)) assuming And(x >= Pi/2, x <= 3*Pi/2)

(2)

Download cosine.mw

Maybe I did something wrong......

Answered: mmcdara 7891

February 10 2025

1 1

... or it's you who either missed something or did something wrong.

test_mmcdara.mw

Is that good?...

Answered: mmcdara 7891

February 08 2025

2 6

>

restart

>

FN := proc(N)
  local C, F, c, nc, T, j:

  C := combinat:-setpartition([$1..N]):
  F := 0:

  for c in C do
    nc := numelems~(c);
    if is({nc[]} subset {1, 2}) then
      T := 1:
      for j in c do
        if numelems(j) = 1 then
          T := T*theta[j[]]
        else
          T := T*B[j]
        end if:
      end do:
      F := F + T
    end if:
  end do:

  return F
end proc:

>	F__2 = FN(2)

(1)

>	F__4 = FN(4)

F__4 = theta[1]*theta[2]*theta[3]*theta[4]+B[[1, 2]]*theta[3]*theta[4]+B[[1, 3]]*theta[2]*theta[4]+B[[1, 4]]*theta[2]*theta[3]+B[[2, 3]]*theta[1]*theta[4]+B[[2, 4]]*theta[1]*theta[3]+B[[3, 4]]*theta[1]*theta[2]+B[[1, 2]]*B[[3, 4]]+B[[1, 3]]*B[[2, 4]]+B[[1, 4]]*B[[2, 3]]

(2)

>	F__6 = FN(6)

(3)

Download Theta_B.mw

A solution...

Answered: mmcdara 7891

February 08 2025

1 0

This nothing but what Statistics:-LinearFit does for numeric points, here generalized to points in abstract form.

The last example leads to an error which could easily be removed while testing in the Parabola procedure if at least two points have the same abscissa but different ordinates

>

restart

>	Parabola := proc(A, B, C) local ABC := `<,>`(A, B, C): local M := `<\|>`(seq(ABC[..,1]^~(i), i=0..2)): y = simplify(`<\|>`(1, x, x^2) . (M^+ . M)^(-1) . (M^+ . ABC[..,2])) end proc:

Generic

>	A := Vector[row](2, symbol=a): B := Vector[row](2, symbol=b): C := Vector[row](2, symbol=c): Parabola(A, B, C)

y = -(x^2*a[1]*b[2]-x^2*a[1]*c[2]-x^2*a[2]*b[1]+x^2*a[2]*c[1]+x^2*b[1]*c[2]-x^2*b[2]*c[1]-x*a[1]^2*b[2]+x*a[1]^2*c[2]+x*a[2]*b[1]^2-x*a[2]*c[1]^2-x*b[1]^2*c[2]+x*b[2]*c[1]^2-a[1]^2*b[1]*c[2]+a[1]^2*b[2]*c[1]+a[1]*b[1]^2*c[2]-a[1]*b[2]*c[1]^2-a[2]*b[1]^2*c[1]+a[2]*b[1]*c[1]^2)/(a[1]^2*b[1]-a[1]^2*c[1]-a[1]*b[1]^2+a[1]*c[1]^2+b[1]^2*c[1]-b[1]*c[1]^2)

(1)

Numeric example

>	A := `<\|>`(-1, 1): B := `<\|>`( 0, 0): C := `<\|>`( 1, 1): Parabola(A, B, C)

(2)

Do we retreive the inital parabola we started from?

Example 1

>	P := x -> Ux^2 + Vx + W; A := Vector[row](2, [ 1, P( 1)]): B := Vector[row](2, [ 2, P( 2)]): C := Vector[row](2, [-1, P(-1)]): Parabola(A, B, C)

(3)

Do we retreive the inital parabola we started from?

Example 2

>	P := x -> 0x^2 + Vx + W; A := Vector[row](2, [ 1, P( 1)]): B := Vector[row](2, [ 2, P( 2)]): C := Vector[row](2, [-1, P(-1)]): Parabola(A, B, C)

(4)

Do we retreive the inital parabola we started from?

Example 3

>	P := x -> 0x^2 + 0x + W; A := Vector[row](2, [ 1, P( 1)]): B := Vector[row](2, [ 2, P( 2)]): C := Vector[row](2, [-1, P(-1)]): Parabola(A, B, C)

(5)

Points aligned along th line y = x

>	P := x -> Ux^2 + Vx + W; A := Vector[row](2, [0, 0]): B := Vector[row](2, [1, 1]): C := Vector[row](2, [2, 2]): Parabola(A, B, C)

(6)

The procedure fails when the three points are aligned on a vertical line x=constant

>	P := x -> Ux^2 + Vx + W; A := Vector[row](2, [0, 1]): B := Vector[row](2, [0, 2]): C := Vector[row](2, [0, 3]): Parabola(A, B, C)

Error, (in rtable/Power) numeric exception: division by zero

Download Parabola.mw

When you know that warnings may occur fo...

Answered: mmcdara 7891

February 08 2025

1 0

When you know that warnings may occur for some specific reason you know, use this remove them

interface(warnlevel=0)

But this might be dangerous when you get "sputious warnings" as they may reveal that something does not work as expected.
So be carefull in using the comman above.

In your case it is very easy to find that the display command is the one which produces these 3 warnings.
The reason comes from the command

dr := draw([e1(color = blue), NorA(color = red), NorB(color = red), NorC(color = red), A(color = red, symbol = solidcircle, symbolsize = 12), B(color = red, symbol = solidcircle, symbolsize = 12), C(color = red, symbol = solidcircle, symbolsize = 12), Oo(color = red, symbol = solidcircle, symbolsize = 12)]);

To see what happens just do

draw(NorA(color = red))

What you get it a simple boxed frame with noting drawn inside.
The reason why becomes clear if you run

detail(NorA)

You see then that the equation of the line NorA contains two names A[1] and A[2] instead of numeric values.

To fix this replace

line(NorA, y - A[2] = a^2*A[2]*(x - A[1])/b^2, [x, y]);
line(NorB, y - B[2] = a^2*B[2]*(x - B[1])/b^2, [x, y]);
line(NorC, y - C[2] = a^2*C[2]*(x - C[1])/b^2, [x, y]);

by

line(NorA, y - coordinates(A)[2] = a^2*coordinates(A)[2]*(x - coordinates(A)[1])/b^2, [x, y]);
line(NorB, y - coordinates(B)[2] = a^2*coordinates(B)[2]*(x - coordinates(B)[1])/b^2, [x, y]);
line(NorC, y - coordinates(C)[2] = a^2*coordinates(C)[2]*(x - coordinates(C)[1])/b^2, [x, y]);

sol.mw

PS: Pourriez-vous avoir un peu plus de respect pour ceux dont vous sollicitez de l'aide et livrer une feuille de travail Maple plutôt qu'un copié-collé issu de (je pense) Mistral ?

Is this useful?...

Answered: mmcdara 7891

February 07 2025

2 4

After your second restart you write N:=2: This is inconsistent with what comes after (obviously i ranges from 1 to 3 and not 1 to 2 in thedefinition of f0).

Once this corrected:

First thing: The Iterator package is not part of my Maple 2015 version, so I used in this workshet a workaround that you can ignore and replace by your own two command lines.

Second thing: I don't understand what your last comment "Change b[1,2]*b[2,3]*b[1,3] to b[1,2,3], etc" does mean. More precisely what does etc mean???

So I interpreted this comment in a way which might not be the correct one

m-lump_2015.mw

Step by step explanation of the errors a...

Answered: mmcdara 7891

February 04 2025

2 0

Before trying to use dsolve blindly verify if your problem is correctly posed.

Read carefully the code after the comment STEP by STEP explanation : it is a kind of "debugging process" to understand where your error comes from and how to fix them ny yourselve.

At the end, once you have understood what I wrote, it will be on you to pose your problem correctly.

>

>	kernelopts(version);

(1)

>

eq1 := diff(f(x), x, x, x)-(diff(f(x), x))^2+theta*Gr+f(x)*(diff(f(x), x, x))-M^2*(diff(f(x), x))-(diff(f(x), x))^2*G[1]-(diff(f(x), x))*j[1] = 0;

diff(diff(diff(f(x), x), x), x)-(diff(f(x), x))^2+theta*Gr+f(x)*(diff(diff(f(x), x), x))-M^2*(diff(f(x), x))-(diff(f(x), x))^2*G[1]-(diff(f(x), x))*j[1] = 0

(1+(4/3)*R)*(diff(diff(theta(x), x), x))/Pr+f(x)*(diff(theta(x), x))+Delta*theta-g[1]*theta*(diff(f(x), x))+N[b]*(diff(theta(x), x))*(diff(phi(x), x))+N[t]*(diff(theta(x), x))^2 = 0

diff(diff(phi(x), x), x)+Sc*f(x)*(diff(phi(x), x))+N[t]*(diff(diff(theta(x), x), x))/N[b] = 0

(2)

>

(3)

>

a1 := [M = .8, Gr = 2.0, G[1] = .5, j[1] = .4, N[t] = .7, N[b] = .1, Sc = .4, R = 1.0, Delta = .5, g[1] = .5, Pr = 2.5, eta = 3];

(4)

>

diff(diff(phi(x), x), x)+.4*f(x)*(diff(phi(x), x))+7.000000000*(diff(diff(theta(x), x), x)) = 0

diff(diff(phi(x), x), x)+.4*f(x)*(diff(phi(x), x))+3.500000000*(diff(diff(theta(x), x), x)) = 0

diff(diff(phi(x), x), x)+.4*f(x)*(diff(phi(x), x))+2.333333333*(diff(diff(theta(x), x), x)) = 0

diff(diff(phi(x), x), x)+.4*f(x)*(diff(phi(x), x))+1.750000000*(diff(diff(theta(x), x), x)) = 0

(5)

STEP by STEP explanation

>	# Functions which appear through derivatives select(has, indets(b1, function), diff)

{diff(diff(phi(x), x), x), diff(diff(theta(x), x), x), diff(phi(x), x), diff(theta(x), x)}

(6)

>	# You have to give 2 BCs for phi and theta # BCs for f are useless # # Example bcs_version_1 := theta(0) = 1, theta(7) = 0, phi(0) = 1, phi(7) = 0; # Let us try this sol_version_1 := dsolve({b1, bcs_version_1}, numeric); # No 'eval' is requiered here

Error, (in dsolve/numeric/bvp/convertsys) the ODE system does not contain derivatives of the unknown function f

>

# What this happen?
#
# b1 contains the abstract function f(x) and Maple does not know how to consider it.
# By default Maple considers f(x) is a function defined through some ODE and asks for
# IC/BC.
#
# But we saw above thos ODE does not exist.
# As you are using dsol/numeric (not dsolve/symbolic) you must provide an algebraic expression
# for f(x)
#
# Example

My_f := x -> 1:

b1_version_2 := eval(b1, f = (x -> My_f(x)));

sol_version_12 := dsolve({b1_version_2, bcs_version_1}, numeric)

diff(diff(phi(x), x), x)+.4*(diff(phi(x), x))+7.000000000*(diff(diff(theta(x), x), x)) = 0

Error, (in dsolve/numeric/bvp/convertsys) unable to convert to an explicit first-order system

>

# Still an error
#
# You have two functions which intervene both through derivatives but you have only
# one ODE.
#
# So, either you give one ofthem an analytic expression... or you introduce a second ODE.
#
# Example (choice 1)

My_theta := x -> cos(x):

b1_version_21 := eval(b1_version_2, theta = (x -> My_theta(x)));

sol_version_121 := dsolve({b1_version_21, bcs_version_1}, numeric);

diff(diff(phi(x), x), x)+.4*(diff(phi(x), x))-7.000000000*cos(x) = 0

Error, (in dsolve/numeric/process_input) invalid specification of initial conditions, got {theta(0) = 1, theta(7) = 0}

>

# Here the error is due to the fact that there exist BC on some function (theta) which no
# longer exist.
# Then remove the and look to what it happens

bcs_version_2 := remove(has, {bcs_version_1}, theta);

sol_version_221 := dsolve({b1_version_21, bcs_version_2[]}, numeric);

# Now you get a result without no more error.

(7)

>	# Example (choice 2) theta_ODE := diff(theta(x), x$2) = -cos(x); sol_version_122 := dsolve({b1_version_21, theta_ODE, bcs_version_1}, numeric);

diff(diff(theta(x), x), x) = -cos(x)

(8)

>

Download sps1_mmcdara.mw

>

>	kernelopts(version);

(1)

>

eq1 := diff(f(x), x, x, x)-(diff(f(x), x))^2+theta*Gr+f(x)*(diff(f(x), x, x))-M^2*(diff(f(x), x))-(diff(f(x), x))^2*G[1]-(diff(f(x), x))*j[1] = 0;

diff(diff(diff(f(x), x), x), x)-(diff(f(x), x))^2+theta*Gr+f(x)*(diff(diff(f(x), x), x))-M^2*(diff(f(x), x))-(diff(f(x), x))^2*G[1]-(diff(f(x), x))*j[1] = 0

(1+(4/3)*R)*(diff(diff(theta(x), x), x))/Pr+f(x)*(diff(theta(x), x))+Delta*theta-g[1]*theta*(diff(f(x), x))+N[b]*(diff(theta(x), x))*(diff(phi(x), x))+N[t]*(diff(theta(x), x))^2 = 0

diff(diff(phi(x), x), x)+Sc*f(x)*(diff(phi(x), x))+N[t]*(diff(diff(theta(x), x), x))/N[b] = 0

(2)

>

(3)

>

a1 := [M = .8, Gr = 2.0, G[1] = .5, j[1] = .4, N[t] = .7, N[b] = .1, Sc = .4, R = 1.0, Delta = .5, g[1] = .5, Pr = 2.5, eta = 3];

(4)

>

diff(diff(phi(x), x), x)+.4*f(x)*(diff(phi(x), x))+7.000000000*(diff(diff(theta(x), x), x)) = 0

diff(diff(phi(x), x), x)+.4*f(x)*(diff(phi(x), x))+3.500000000*(diff(diff(theta(x), x), x)) = 0

diff(diff(phi(x), x), x)+.4*f(x)*(diff(phi(x), x))+2.333333333*(diff(diff(theta(x), x), x)) = 0

diff(diff(phi(x), x), x)+.4*f(x)*(diff(phi(x), x))+1.750000000*(diff(diff(theta(x), x), x)) = 0

(5)

STEP by STEP explanation

>	# Functions which appear through derivatives select(has, indets(b1, function), diff)

{diff(diff(phi(x), x), x), diff(diff(theta(x), x), x), diff(phi(x), x), diff(theta(x), x)}

(6)

>	# You have to give 2 BCs for phi and theta # BCs for f are useless # # Example bcs_version_1 := theta(0) = 1, theta(7) = 0, phi(0) = 1, phi(7) = 0; # Let us try this sol_version_1 := dsolve({b1, bcs_version_1}, numeric); # No 'eval' is requiered here

Error, (in dsolve/numeric/bvp/convertsys) the ODE system does not contain derivatives of the unknown function f

>

# What this happen?
#
# b1 contains the abstract function f(x) and Maple does not know how to consider it.
# By default Maple considers f(x) is a function defined through some ODE and asks for
# IC/BC.
#
# But we saw above thos ODE does not exist.
# As you are using dsol/numeric (not dsolve/symbolic) you must provide an algebraic expression
# for f(x)
#
# Example

My_f := x -> 1:

b1_version_2 := eval(b1, f = (x -> My_f(x)));

sol_version_12 := dsolve({b1_version_2, bcs_version_1}, numeric)

diff(diff(phi(x), x), x)+.4*(diff(phi(x), x))+7.000000000*(diff(diff(theta(x), x), x)) = 0

Error, (in dsolve/numeric/bvp/convertsys) unable to convert to an explicit first-order system

>

# Still an error
#
# You have two functions which intervene both through derivatives but you have only
# one ODE.
#
# So, either you give one ofthem an analytic expression... or you introduce a second ODE.
#
# Example (choice 1)

My_theta := x -> cos(x):

b1_version_21 := eval(b1_version_2, theta = (x -> My_theta(x)));

sol_version_121 := dsolve({b1_version_21, bcs_version_1}, numeric);

diff(diff(phi(x), x), x)+.4*(diff(phi(x), x))-7.000000000*cos(x) = 0

Error, (in dsolve/numeric/process_input) invalid specification of initial conditions, got {theta(0) = 1, theta(7) = 0}

>

# Here the error is due to the fact that there exist BC on some function (theta) which no
# longer exist.
# Then remove the and look to what it happens

bcs_version_2 := remove(has, {bcs_version_1}, theta);

sol_version_221 := dsolve({b1_version_21, bcs_version_2[]}, numeric);

# Now you get a result without no more error.

(7)

>	# Example (choice 2) theta_ODE := diff(theta(x), x$2) = -cos(x); sol_version_122 := dsolve({b1_version_21, theta_ODE, bcs_version_1}, numeric);

diff(diff(theta(x), x), x) = -cos(x)

(8)

Download sps1_mmcdara.mw

Good point...

Answered: mmcdara 7891

February 03 2025

1 3

EDITED Wed 5th Feb 2025 at 6:50 UT

@segfault

@vv : Thank you for having pointed out this mistake.

@segfault : Here is a modified procedure which seems correct: Download Into_Sine.mw

UPDATED : Here is a version which gives more concise expressions
Into_Sine_more_concise.mw

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