The convolution of f(x,y) and g(x,y) is defined by (f*g)(x,y), so let f(x,y) be

Question

The convolution of f(x,y) and g(x,y) is defined by (f*g)(x,y), so let f(x,y) be a prescribed function of two variables and define g(x,y;t)

a) For each t, find g(x,y;t)dxdy

b) Show that u(x,y,0)=(f*g(;t))(x,y;t) solves u_t=u_xx+u_yy

c) What is u(x,y,0)?

The convolution of f(x,y) and g(x,y) is defined by (f*g)(x,y), so let f(x,y) be a prescribed function of two variables and define g(x,y;t) Henceforth, let f(x,y) be a prescribed function of two variables and define g(x,y;t) = 1/4pit e^-(x^2 + y^2)/4t a) For each t, find double integrate g(x,y;t)dxdy b) Show that u(x,y,0)=(f*g(;t))(x,y;t) solves u_t=u_xx+u_yy c) What is u(x,y,0)?Explain your answer briefly

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