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pdetest Not Giving 0 for pdsolve...

Test007 50 Maple 2020
pde pdsolve differential-equations
July 26 2020
1 1

Does anyone know why pdetest is not giving [0, 0, 0] for the solution of pdsolve?
 

> 

restart
> 

eq := diff(u(x, t), t)-k*(diff(u(x, t), x, x)) = 0

diff(u(x, t), t)-k*(diff(diff(u(x, t), x), x)) = 0

 (1)
> 

ic := u(x, 0) = g(x)

u(x, 0) = g(x)

 (2)
> 

bc := (D[1](u))(0, t) = 0, eval(diff(u(x, t), x)+u(x, t), x = 1) = 0

(D[1](u))(0, t) = 0, eval(diff(u(x, t), x), {x = 1})+u(1, t) = 0

 (3)
> 

sol := `assuming`([simplify(pdsolve([eq, ic, bc]))], [k > 0, 0 < x and x < 1, t > 0])

u(x, t) = `casesplit/ans`(Sum(4*exp(-k*lambda[n]^2*t)*lambda[n]*cos(lambda[n]*x)*(Int(g(x)*cos(lambda[n]*x), x = 0 .. 1))/(2*lambda[n]+sin(2*lambda[n])), n = 0 .. infinity), {And((sin(lambda[n])*lambda[n]-cos(lambda[n]))/cos(lambda[n]) = 0, 0 < lambda[n])})

 (4)
> 

`assuming`([simplify(pdetest(sol, [eq, ic, bc]))], [k > 0, 0 < x and x < 1, t > 0])

[-(diff(diff(`casesplit/ans`(sum(4*exp(-k*lambda[n]^2*t)*lambda[n]*cos(lambda[n]*x)*(Int(g(x)*cos(lambda[n]*x), x = 0 .. 1))/(2*lambda[n]+sin(2*lambda[n])), n = 0 .. infinity), {And(-(1/2)*(I*lambda[n]*exp(I*lambda[n])-I*lambda[n]*exp(-I*lambda[n])+exp(I*lambda[n])+exp(-I*lambda[n]))/cos(lambda[n]) = 0, 0 < lambda[n])}), x), x))*k+diff(`casesplit/ans`(sum(4*exp(-k*lambda[n]^2*t)*lambda[n]*cos(lambda[n]*x)*(Int(g(x)*cos(lambda[n]*x), x = 0 .. 1))/(2*lambda[n]+sin(2*lambda[n])), n = 0 .. infinity), {And(-(1/2)*(I*lambda[n]*exp(I*lambda[n])-I*lambda[n]*exp(-I*lambda[n])+exp(I*lambda[n])+exp(-I*lambda[n]))/cos(lambda[n]) = 0, 0 < lambda[n])}), t), `casesplit/ans`(Sum(4*lambda[n]*cos(lambda[n]*x)*(Int(g(x)*cos(lambda[n]*x), x = 0 .. 1))/(2*lambda[n]+sin(2*lambda[n])), n = 0 .. infinity), {And(-(1/2)*(I*lambda[n]*exp(I*lambda[n])-I*lambda[n]*exp(-I*lambda[n])+exp(I*lambda[n])+exp(-I*lambda[n]))/cos(lambda[n]) = 0, 0 < lambda[n])})-g(x), eval(diff(`casesplit/ans`(Sum(4*exp(-k*lambda[n]^2*t)*lambda[n]*cos(lambda[n]*x)*(Int(g(x)*cos(lambda[n]*x), x = 0 .. 1))/(2*lambda[n]+sin(2*lambda[n])), n = 0 .. infinity), {And(-(1/2)*(I*lambda[n]*exp(I*lambda[n])-I*lambda[n]*exp(-I*lambda[n])+exp(I*lambda[n])+exp(-I*lambda[n]))/cos(lambda[n]) = 0, 0 < lambda[n])}), x), {x = 0}), eval(diff(`casesplit/ans`(Sum(4*exp(-k*lambda[n]^2*t)*lambda[n]*cos(lambda[n]*x)*(Int(g(x)*cos(lambda[n]*x), x = 0 .. 1))/(2*lambda[n]+sin(2*lambda[n])), n = 0 .. infinity), {And(-(1/2)*(I*lambda[n]*exp(I*lambda[n])-I*lambda[n]*exp(-I*lambda[n])+exp(I*lambda[n])+exp(-I*lambda[n]))/cos(lambda[n]) = 0, 0 < lambda[n])}), x), {x = 1})+`casesplit/ans`(Sum(4*exp(-k*lambda[n]^2*t)*lambda[n]*cos(lambda[n])*(Int(g(x)*cos(lambda[n]*x), x = 0 .. 1))/(2*lambda[n]+sin(2*lambda[n])), n = 0 .. infinity), {And(-(1/2)*(I*lambda[n]*exp(I*lambda[n])-I*lambda[n]*exp(-I*lambda[n])+exp(I*lambda[n])+exp(-I*lambda[n]))/cos(lambda[n]) = 0, 0 < lambda[n])})]

 (5)
> 

``


 

Download PDE.mw

Change Summation Variable in OrthogonalE...

Test007 50 Maple 2020
substitution summation orthogonalexpansions
July 18 2020
1 3

In the OrthogonalExpansions package, how can I change the summation variable to be n instead of i?

General Solution of ODE with Arbitrary C...

Test007 50 Maple 2020
ode dsolve differential-equations
July 10 2020
2 5

The general solution of x2y'' + 3xy' + λy = 0 is (from Example 4 here)

Why does dsolve(x^2*diff(y(x), x, x) + 3*x*diff(y(x), x) + lambda*y(x) = 0) give only the third case instead of the whole solution?

Incorrect Result from pdsolve for Diffus...

Test007 50 Maple 2020
pde pdsolve differential-equations
June 12 2020
1 0

I want to solve the following IBVP:

However, the answer given by pdsolve doesn't give  with pdetest.


 

> 

restart
> 

eq := diff(u(x, t), t)-k*(diff(u(x, t), x, x)) = 0

diff(u(x, t), t)-k*(diff(diff(u(x, t), x), x)) = 0

 (1)
> 

ic := u(x, 0) = 0

u(x, 0) = 0

 (2)
> 

bc := u(0, t) = p(t)

u(0, t) = p(t)

 (3)
> 

sol := `assuming`([simplify(pdsolve([eq, ic, bc]))], [k > 0, x > 0, t > 0])

u(x, t) = (1/2)*x*(Int(p(zeta)*exp(-(1/4)*x^2/(k*(t-zeta)))/(t-zeta)^(3/2), zeta = 0 .. t))/(Pi^(1/2)*k^(1/2))

 (4)
> 

`assuming`([simplify(pdetest(sol, [eq, ic, bc]))], [k > 0, t > 0, x > 0])

[(1/2)*x*(-t*(Int(p(zeta)*exp(-(1/4)*x^2/(k*(t-zeta)))/(t-zeta)^(7/2), zeta = 0 .. t))+limit(p(zeta)*exp(-(1/4)*x^2/(k*(t-zeta)))/(t-zeta)^(3/2), zeta = t)+Int(p(zeta)*exp(-(1/4)*x^2/(k*(t-zeta)))*zeta/(t-zeta)^(7/2), zeta = 0 .. t)+Int(p(zeta)*exp(-(1/4)*x^2/(k*(t-zeta)))/(t-zeta)^(5/2), zeta = 0 .. t))/(Pi^(1/2)*k^(1/2)), 0, -p(t)]

 (5)
> 

``

Download IBVP.mw

Why Is Inhomogeneous Heat Equation Formu...

Test007 50 Maple 2020
pde heat differential-equations
June 06 2020
1 6

From Wikipedia,

However, when I plug the formula of u(x, t) into Maple, it doesn't seem to satisfy the PDE and is stuck evaluating.
 

> 

restart
> 

eq := diff(u(x, t), t)-k*(diff(u(x, t), x, x)) = f(x, t)

diff(u(x, t), t)-k*(diff(diff(u(x, t), x), x)) = f(x, t)

 (1)
> 

ic := u(x, 0) = 0

u(x, 0) = 0

 (2)
> 

"G(x,t):=1/(sqrt(4*pi*k*t))exp(-(x^(2))/(4*k*t))"

proc (x, t) options operator, arrow, function_assign; exp(-(1/4)*x^2/(k*t))/sqrt(4*pi*k*t) end proc

 (3)
> 

ans := u(x, t) = int(G(x-y, t-s)*f(y, s), y = -infinity .. infinity, s = 0 .. t)

u(x, t) = (1/2)*(int((int(exp(-(1/4)*(x-y)^2/(k*(t-s)))*f(y, s), y = -infinity .. infinity))/(pi*k*(t-s))^(1/2), s = 0 .. t))

 (4)
> 

`assuming`([simplify(pdetest(ans, [eq, ic]))], [t > 0])
> 

``


 

Download InhomoHeat.mw

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