Answer to Question #187340 in Differential Equations for irfan

Question

Solve IBVP

∂u/∂t (x,t)=2 (∂^2 u)/(∂x^2 )+x+2,0<x<1,t>0

u(x,0)=x+2,0<x<1,u(0,t)=2,u(2,t)=-2

Answer 1

$Given pde : \frac{\partial u}{\partial t} =2 \frac{\partial^2 u}{\partial x^2}+x+2, 0<x<1,t>0\newline u(x,0)=x+2,0<x<1\newline u(0,t)=2\newline u(2,t)=-2\newline \text{The given pde is non homogeneous heat equation.}\newline$$\text{Now, let }u(x,t)=v(x,t)+Ax+B\newline \text{Therefore, } \frac{\partial u}{\partial t} =\frac{\partial v}{\partial t} ,\space \frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 v}{\partial x^2}\newline \text{We get, A=B=2.} Then,\newline v(x,0)=x,0<x<1\newline v(0,t)=0\newline v(2,t)=0\newline \text{Therefore, the solution is given by}\newline v(x,t)=\sum_{n=1}^\infty E_{n}(x)sin(\frac{n\pi x}{l}) e^{\frac{- \alpha n ^2 \pi ^2 t }{l^2}}+\sum_{n=1}^\infty sin(\frac{n\pi x}{l}) \int_{0} ^{t} e^{\frac{- \alpha n ^2 \pi ^2 (t-s) }{l^2}}F_{n}(s)ds\newline \text{Where} \space E_{n}(x)=\frac{2}{l} \int_{0} ^{l} F(x)sin(\frac{n\pi x}{l}) dx,\newline F_{n}(s)=\frac{2}{l} \int_{0} ^{t} \phi (s,t)sin(\frac{n\pi s}{l}) ds\newline \text{Here, we have} \newline F(x)=x,\phi (x,t)=x+2, \alpha=2,l=2.\newline \text{Therefore, } E_{n}(x)=\int_{0} ^{2} xsin(\frac{n\pi x}{2}) dx\newline =\frac{(-1)^{n+1}4}{n\pi }\newline F_{n}(s)=\int_{0} ^{\infty} (s+2)sin(\frac{n\pi s}{2}) ds\newline \text{Therefore, solution is} \newline u(x,t)=\sum_{n=1}^\infty E_{n}(x)sin(\frac{n\pi x}{l}) e^{\frac{- \alpha n ^2 \pi ^2 t }{l^2}}+\sum_{n=1}^\infty sin(\frac{n\pi x}{l}) \int_{0} ^{t} e^{\frac{- \alpha n ^2 \pi ^2 (t-s) }{l^2}}F_{n}(s)ds+2x+2\newline where \space E_{n}(x)=\frac{(-1)^{n+1}4}{n\pi }\newline F_{n}(s)=\int_{0} ^{\infty} (s+2)sin(\frac{n\pi s}{2}) ds$