8E: Excercise

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Nov 20, 2020
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- 8.3: Wave Equations
- 9: Vector Calculus

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Exercise $\PageIndex{1}$ : Heat Equation

Let $u(x,t)= t^{-1/2} e^{-x^2/4t}.$ Find $\frac{\partial ^2u}{\partial x^2}-\frac{\partial u}{\partial t}.$
Let $u(x,y,t)= t^{-1} e^{-(x^2+y^2)/4t}.$ Find $\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2}-\frac{\partial u}{\partial t}.$

Exercise $\PageIndex{2}$ : Laplace equation

Show that function $z(x, y) = 2e^{\pi x} sin(\pi y)$ satisfies the two dimensional Laplace equation:

$\frac{\partial ^2z}{\partial x^2}+\frac{\partial ^2z}{\partial y^2}=0.$

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Exercise 8E.1\PageIndex{1}: Heat Equation

Exercise 8E.2\PageIndex{2}: Laplace equation

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Exercise $\PageIndex{1}$ : Heat Equation

Exercise $\PageIndex{2}$ : Laplace equation