Invariance properties for the diffusion equation. (a) Show that if u(x, t) is a solution of...

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Invariance properties for the diffusion equation. (a) Show that if u(x, t) is a solution of the diffusion equation on the whole line, then so is any derivative of u, e.g., u(x, t) or u(x, t). (b) Show that if u(x, t) is a solution, then so is u(√ax, at) for each a > 0. (c) Show that if u(x, t) is a solution of the diffusion equation on the whole line, then so is its convolution with a function f(x), v(x, t) = (u(., t) * f) (x):= = u(x - y, t) f (y) dy. You may assume that the integral converges. Hint: Pass the derivatives inside the integral. Invariance properties for the diffusion equation. (a) Show that if u(x, t) is a solution of the diffusion equation on the whole line, then so is any derivative of u, e.g., u(x, t) or u(x, t). (b) Show that if u(x, t) is a solution, then so is u(√ax, at) for each a > 0. (c) Show that if u(x, t) is a solution of the diffusion equation on the whole line, then so is its convolution with a function f(x), v(x, t) = (u(., t) * f) (x):= = u(x - y, t) f (y) dy. You may assume that the integral converges. Hint: Pass the derivatives inside the integral.