Question

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5. Propagation speed. (a) How do you know that the heat equation u kuzz implies infinite propagation speed? (b) How do you know that the wave equation utt-t2uzz İmplies propagation with the speed c ?

Answer 1

Given data (a) u t = x u_x x Let psi: R rightarrow R be bounded define u (x, t) = integral_IR k_1 (x, y, z) psi (r) dy = 1/squareroot 8 pi t integral_IR cx p(- (x - q)^q/4t psi (g) dy Then u (x, t) is the solution Now this solution of any point (x, t) elementof R times (0, alpha) depends on the values of vertical data psi (x) at all x elementof IR, then the domain of plane is (at any (x, t) for t > 0) in the real line (-alpha, alpha) Moreover, clearly it reaches all points shortly it is suitable for physical cases Heat equation describes and analyze the hall parts of any quantity that different an a consequences of spatial gradients in its concentration