Word Count : 595

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We are asked to solve the wave equation [ u_{tt} = u_{xx}, \quad 0 < x < 1, \ t>0 ] with [ u(x,0) = 0, \quad u_t(x,0) = 0, \quad u(0,t) = 0, \quad u(1,t) = 200 \sin(\pi t), ] using the explicit finite difference method with (h = k = 0.25) up to four time levels. --- Step 1: Discretization Let (x_i = i h, \ i=0,1,2,3,4) (since (h=0.25) and (x\in[0,1])) and (t^n = n k, \ n=0,1,2,3,4). Let (u_i^n \approx u(x_i,t^n)). The explicit scheme for the wave equation is: [ u_i^{n+1} = 2(1-r^2) u_i^n - u_i^{n-1} + r^2 (u_{i+1}^n + u_{i-1}^n), \quad r = \frac{k}{h}. ] Here (k=h=0.25 \implies r = \frac{0.25}{0.25} = 1). So the scheme becomes: [ u_i^{n+1} = u_{i+1}^n + u_{i-1}^n - u_i^{n-1}. ] Step 2: Grid points * (x_0=0, x_1=0.25, x_2=0.5, x_3=0.75, x_4=1) * Boundary conditions: [ u_0^n = 0, \quad _______ __________ ______ ____ _____ ____ ___ _____ ________ __________.
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