# Partial Differential Equations

1. Consider the first order PDE,

∂u/∂t = ct(∂u/∂x) -∞ < x < ∞ c does not equal 0

a) Find the fundamental solution

b) Use the fundamental solution and convolution to find a formula for the solution to:

∂u/∂t = ct(∂u/∂x) -∞ < x < ∞ c does not equal 0 u(x,0) = f(x)

c) Use the method of characteristics to find the solution to part b.(Make sure your answers agree)

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1. Consider the first order PDE,

∂u/∂t = ct(∂u/∂x) -∞ < x < ∞ c does not equal 0

a) Find the fundamental solution

b) Use the fundamental solution and convolution to find a formula for ...

#### Solution Summary

This finds the fundamental solution of a first-order partial differential equation, and uses that solution and convolution, as well as method of characteristics, in another problem. The expert uses the method of characteristics to find the function.

Partial Differential Equations : Heat Equations

1) Let A(x,y) be the area of a rectangle not degenerated of dimensions x and y, in a way that the rectangle is inside a circle of a radius of 10. Determine the domain and the range of this function.

2) The wave equation (c^2 ∂^2 u / ∂ x^2 = ∂^2 u / ∂ t^2) and the heat equation (c ∂^2 u / ∂ x^2 = ∂ u / ∂ t) are two of the most important equations of physics (c is a constant). They are called partial differential equations. Show the following:

a) u = cos x cos ct and u = e^x cosh ct satisfies the wave equation.

b) u = e^-ct sin x and u = t^-1/2 e^[(-x^2)/(4ct)] satisfies the heat equation.

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