# Heat Equation 3-D

I'm new to using Mathematica software, and I need to solve a problem involving the Heat Equation for a quadrangular (3D) prism. I have the equations below and I need the temperature x time graphs; Temperature distribution on faces x, y, z in a fixed time. Is it possible to solve in Mathematica?

• Have you tried anything? Where are you stuck? – Kuba Jun 14 '17 at 11:27
• I'm trying the solution provided by the friend below. But the software is not calculating the Anmp right. The solution should be a cosine-only function, but it is resulting in a cosine function for the first integral and sines for the last two. – Kevin Sacramento Vivas Neres Jun 14 '17 at 12:37
• Can they be converted into cosine functions? Could be MMA is just doing some cleaning up that you don't want. You could try TrigReduce@Anmp[n,m,p] which gives me an answer that's much nastier, but all cosines. – N.J.Evans Jun 14 '17 at 14:14

Here is a starting for you,

u0 = 25; L1 = 1; L2 = 1; L3 = 2; alpha = 79.90*10^(-6);

Anmp[n_, m_, p_] =
8*u0/(L1*L2*L3)*
Integrate[Sin[n*Pi/L1*x]*Sin[m*Pi/L2*y]*Sin[p*Pi/L3*z], {z, 0, L3}, {y, 0, L2}, {x, 0, L1}]

u[x_, y_, z_, t_] =
Sum[Sum[Sum[Anmp[n, m, p]*Sin[n*Pi/L1*x]*Sin[m*Pi/L2*y]*Sin[p*Pi/L3*z]*
Exp[-alpha*((n*Pi/L1)^2 + (m*Pi/L2)^2 + (p*Pi/L3)^2)*t], {p, 0,
Infinity}], {m, 0, Infinity}], {n, 0, Infinity}]


Try with ContourPlot3D.

• The answer to the ANMP should only be a cosine function, but the software is returning a cosine and sine function. – Kevin Sacramento Vivas Neres Jun 14 '17 at 12:22
• u0 = 25; L1 = 1; L2 = 1; L3 = 2; alpha = 79.90*10^(-6); Anmp[n_, m_, p_] = 8*u0/(L1*L2*L3)*Integrate[Sin[nPix/L1]*Sin[mPiy/L2]*Sin[pPiz/L3], {z, 0, L3}, {y, 0, L2}, {x, 0, L1}] – Kevin Sacramento Vivas Neres Jun 14 '17 at 12:29
• Only one correction in the position of x, y, z – Kevin Sacramento Vivas Neres Jun 14 '17 at 12:30