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Question: pdsolve solution with unexpected imaginary term

Question:pdsolve solution with unexpected imaginary term

Scot Gould 872 Maple 2024
pde pdsolve differential-equations
October 18 2024
3

2D Heat Equation problem.

 

In most cases, pdsolve generates expected results. For this one, it does not. Is there an obvious error

in my worksheet?

 

> 

restart; interface(imaginaryunit = i); heat_equation := diff(T(x, y, t), t) = k*(diff(T(x, y, t), x, x)+diff(T(x, y, t), y, y))

 

Writing out the boundary conditions:

> 

bces := T(0, y, t) = 0, T(L, y, t) = 0, T(x, 0, t) = 0, T(x, L, t) = 0

Writing out the initial conditions

> 

ice := T(x, y, 0) = T__0(x, y)

 

where

> 

"`T__0`(x,y) := 1/(4)*sin((2 Pi*x)/(L) )^(2)*sin((3 Pi*y)/(L)):"

 

Showing the initial condition of the system.

> 

L := 1; plot3d_display := scaling = constrained, size = [600, 600], orientation = [-121, 75, 1]; plot3d(T__0(x, y), x = 0 .. L, y = 0 .. L, plot3d_display)

 

Solving the heat equation.

> 

sols := `assuming`([pdsolve({bces, ice, heat_equation}, T(x, y, t))], [k > 0])

T(x, y, t) = Sum(Sum(piecewise(n = 4, -((1/8)*I)*sin(n1*Pi*y)*sin(4*Pi*x)*exp(-Pi^2*k*t*(n1^2+16)), -4*sin(n*Pi*x)*exp(-Pi^2*k*t*(n^2+n1^2))*(-1+(-1)^n)*sin(n1*Pi*y)/(Pi*(n^3-16*n))), n = 1 .. infinity), n1 = 1 .. infinity)

 (1)

Notice the imaginary expression in the solution?

 

> 

Tsol := value(eval(rhs(sols), infinity = 12))

 

Looking at one value that includes an imaginary term.

> 

eval(Tsol, {k = .1, t = 0, x = .2, y = .2})

-.3490991334-.1130635622*I

 (2)

And plotting at t = 0.

> 

plot3d(eval(Tsol, {k = .3, t = 0}), x = 0 .. L, y = 0 .. L, plot3d_display)

 
> 

NULL

Note, if the sin(3*Pi*y/L) term is squared, then pdsolve returns a real solution.

NULL

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