### 8 Heat PDE in rectangle

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#### 8.1 4 edges are ﬁxed at zero temperature, No source

problem number 72

Taken from Maple help pages on PDE

Solve the heat equation for $$u(x,y,t)$$

$\frac{ \partial u}{\partial t}= 1/10 \left ( \frac{ \partial ^2 u}{\partial x^2} + \frac{ \partial ^2 u}{\partial y^2} \right )$

For $$0<x<1$$ and $$0<y<1$$ and $$t>0$$. The boundary conditions are \begin{align*} u(0,y,t) &= 0 \\ u(1,y,t) &= 0 \\ u(x,0,t) &= 0 \\ u(x,1,t) &= 0 \end{align*}

Initial condition is $$u(x,y,0)=x(1-x)(1-y)y$$.

Mathematica

$\text{DSolve}\left [\left \{u^{(0,0,1)}(x,y,t)=\frac{1}{10} \left (u^{(0,2,0)}(x,y,t)+u^{(2,0,0)}(x,y,t)\right ),u(x,y,0)=(1-x) x (1-y) y,\{u(0,y,t)=0,u(1,y,t)=0,u(x,0,t)=0,u(x,1,t)=0\}\right \},u(x,y,t),\{x,y,t\}\right ]$

Maple

$u \left ( x,y,t \right ) =\sum _{{\it n1}=1}^{\infty } \left ( \sum _{n=1}^{\infty }-16\,{\frac{ \left ( - \left ( -1 \right ) ^{{\it n1}+n}+ \left ( -1 \right ) ^{n}+ \left ( -1 \right ) ^{{\it n1}}-1 \right ) \sin \left ( n\pi \,x \right ) \sin \left ({\it n1}\,\pi \,y \right ){{\rm e}^{-1/10\,{\pi }^{2}t \left ({n}^{2}+{{\it n1}}^{2} \right ) }}}{{n}^{3}{\pi }^{6}{{\it n1}}^{3}}} \right )$

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#### 8.2 2 edges are ﬁxed at zero temperature, 2 edges are insulated, internal source term

problem number 73

Taken from Maple help pages on PDE

Solve the heat equation for $$u(x,y,t)$$

$\frac{ \partial u}{\partial t}= 1/10 \left ( \frac{ \partial ^2 u}{\partial x^2} + \frac{ \partial ^2 u}{\partial y^2} \right ) -\frac{1}{5} u(x,y,t);$

For $$0<x<1$$ and $$0<y<1$$ and $$t>0$$. The boundary conditions are \begin{align*} \frac{\partial u}{\partial x} u(0,y,t) &= 0 \\ u(1,y,t) &= 0 \\ u(x,0,t) &= 0 \\ \frac{\partial u}{\partial y} u(x,1,t) &= 0 \\ \end{align*}

Initial condition is $$u(x,y,0)=(1-x^2)(1- \frac{1}{2} y) y$$.

Mathematica

$\text{DSolve}\left [\left \{u^{(0,0,1)}(x,y,t)=\frac{1}{10} \left (u^{(0,2,0)}(x,y,t)+u^{(2,0,0)}(x,y,t)\right )-\frac{1}{5} u(x,y,t),u(x,y,0)=\left (1-x^2\right ) \left (1-\frac{y}{2}\right ) y,\left \{u^{(1,0,0)}(0,y,t)=0,u(1,y,t)=0,u(x,0,t)=0,u^{(0,1,0)}(x,1,t)=0\right \}\right \},u(x,y,t),\{x,y,t\}\right ]$

Maple

$u \left ( x,y,t \right ) =\sum _{{\it n1}=0}^{\infty } \left ( \sum _{n=0}^{\infty }512\,{\frac{ \left ( -1 \right ) ^{n}{{\rm e}^{-1/10\,t \left ( 2+ \left ({n}^{2}+{{\it n1}}^{2}+n+{\it n1}+1/2 \right ){\pi }^{2} \right ) }}\sin \left ( 1/2\, \left ( 1+2\,{\it n1} \right ) \pi \,y \right ) \cos \left ( 1/2\, \left ( 1+2\,n \right ) \pi \,x \right ) }{{\pi }^{6} \left ( 1+2\,n \right ) ^{3} \left ( 1+2\,{\it n1} \right ) ^{3}}} \right )$

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#### 8.3 Rectangle plate, losing heat by convection (Articolo 6.6.3)

problem number 74

Added December 20, 2018.

Example 6.6.3 from Partial diﬀerential equations and boundary value problems with Maple/George A. Articolo, 2nd ed :

We seek the temperature distribution in a thin rectangular plate over the ﬁnite two-dimensional domain $$D ={(x, y) \text{s.t.} 0<x<1, 0<y<1}$$. The lateral surfaces of the plate are insulated. The boundaries $$y = 0$$ and $$y = 1$$ are ﬁxed at temperature $$0$$, the boundary $$x = 0$$ is insulated, and the boundary $$x = 1$$ is losing heat by convection into a surrounding medium at temperature $$0$$. The initial temperature distribution f(x, y) is $u(x,y,0) = \left (1- \frac{x^2}{3} \right ) y(1-y)$

The thermal diﬀusivity is $$k = \frac{1}{50}$$.

Solve for $$u(x,y,t)$$ the heat PDE

$\frac{ \partial u}{\partial t}= k \left ( \frac{ \partial ^2 u}{\partial x^2} +\frac{ \partial ^2 u}{\partial y^2} \right )$

With $$0<x<1,0<y<1$$ and $$t>0$$.

With boundary conditions \begin{align*} \frac{\partial u}{\partial x}(0,y,t) &= 0 \\ \frac{\partial u}{\partial x}(1,y,t) + u(1,y,t) &= 0 \\ u(x,0,t) &= 0\\ u(x,1,t) &=0 \end{align*}

Mathematica

$\left \{\left \{u(r,t)\to \underset{n=1}{\overset{\infty }{\sum }}\frac{2 e^{-t \text{BesselJZero}(0,n)^2} \text{BesselJ}(0,r \text{BesselJZero}(0,n)) \left (\frac{\text{BesselJ}(1,\text{BesselJZero}(0,n))}{\text{BesselJZero}(0,n)}-\frac{1}{3} \text{HypergeometricPFQ}\left (\left \{\frac{3}{2}\right \},\left \{1,\frac{5}{2}\right \},-\frac{1}{4} \text{BesselJZero}(0,n)^2\right )\right )}{\text{BesselJ}(0,\text{BesselJZero}(0,n))^2+\text{BesselJ}(1,\text{BesselJZero}(0,n))^2}\right \}\right \}$

Maple

$u \left ( x,y,t \right ) =\mbox{{\tt casesplit/ans}} \left ( \sum _{n=1}^{\infty } \left ( \sum _{{\it n1}=0}^{\infty }{\frac{32\,{{\rm e}^{-{\frac{1}{50}}\,t \left ({\pi }^{2}{n}^{2}+{\lambda _{{{\it n1}}}}^{2} \right ) }} \left ( \left ( -1 \right ) ^{n}-1 \right ) \cos \left ( \lambda _{{{\it n1}}}x \right ) \sin \left ( n\pi \,y \right ) \left ( -{\lambda _{{{\it n1}}}}^{2}\sin \left ( \lambda _{{{\it n1}}} \right ) +\lambda _{{{\it n1}}}\cos \left ( \lambda _{{{\it n1}}} \right ) -\sin \left ( \lambda _{{{\it n1}}} \right ) \right ) }{3\,{\pi }^{3}{n}^{3}{\lambda _{{{\it n1}}}}^{2} \left ( \sin \left ( 2\,\lambda _{{{\it n1}}} \right ) +2\,\lambda _{{{\it n1}}} \right ) }} \right ) , \left \{{\it And} \left ( \tan \left ( \lambda _{{{\it n1}}} \right ) \lambda _{{{\it n1}}}-1=0,0<\lambda _{{{\it n1}}} \right ) \right \} \right )$