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(d) What is the main difference between the wave equation and the heat equation in terms of the speed of propagation of information?

(d) What is the main difference between the wave equation and the heat equation in terms of the speed of propagation of information?

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part D

(a) D'Alembert's formula for the solution of the wave equation on R is given by 1 rx+ct U(x,t) -(* + ct) + f(x - ct) + 8()ds. %3D Provide a brief discussion of the meaning of the two terms in the right-hand side of the above formula. (b) Show by direct computation that D'Alembert's formula is a solution to the problem Un - eux = 0, U(x,0) = f(x), U(x,0) = g(x). xE R, t> 0, (c) Find the solution to the problem Un - cU = 0, U(x,0) = 0, U;(x,0) x E R, t > 0, = COS X. (d) What is the main difference between the wave equation and the heat equation in terms of the speed of propagation of information?
Transcribed Image Text:(a) D'Alembert's formula for the solution of the wave equation on R is given by 1 rx+ct U(x,t) -(* + ct) + f(x - ct) + 8()ds. %3D Provide a brief discussion of the meaning of the two terms in the right-hand side of the above formula. (b) Show by direct computation that D'Alembert's formula is a solution to the problem Un - eux = 0, U(x,0) = f(x), U(x,0) = g(x). xE R, t> 0, (c) Find the solution to the problem Un - cU = 0, U(x,0) = 0, U;(x,0) x E R, t > 0, = COS X. (d) What is the main difference between the wave equation and the heat equation in terms of the speed of propagation of information?
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