1. Find the solution of the wave equation a² u Ət² [ a² u . c² მ2 მy2 + 0 < x <1, 0 < y < 1 with the boundary conditions u = 0, x = 0,1 ди = 0, y = 0,1 მყ and the initial conditions u(x, y, 0) = x(1 − x)y and ut(x, y, 0) = 0.
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- a2u satisfies the wave equation əx² -n a²u Verify that U(x, t) = e¬Vkt cos\ax %3D k at24. Consider a wave equation on an infinite line, J²u J²u 9 Ət² əx² = 0. = Find the characteristics though the point (1,3). Draw the domains of depen- dence and influence of the point (1,3).Consider the wave equation 0 0, with u(0,1) = 1(xt)= 0, u(x,0) = sin x and =0 at t=0. Then u is
- 3. Prove that the general solution of the wave equation ut = curr is u(x, t) = F(x + ct) +G(x-ct), where ¿2²uxx F and G are c¹ functions. (Hint: let & = x+ct and n = x-ct) Utt= Use variables separation method to solve the wave equation uxxutt. This function is defined on spatial domain 0 0. Subject to boundary conditions: ux(0, t) = u,(a, t) = 0 and initial conditions: u(x, 0) = 0 and u₁(x,0) = f(x)5C. Under suitable assumptions derive one dimensional wave equation.
- Q2) Solve the wave equation u=u 00 " XX 11 subject to u(0,t) = u(2,t) =0, t>0 and u(x,0)=0, u )=0, u,(x,0): = sin (3лx), 08) Find the position vector r(t) for a particle with acceleration a(t) = (5t, 5 sin t, cos 6t), initial velocity (0) = (3, -3, 1) and initial position (0) = (5, 0, -2).Consider the wave equation utt = uxx, (x, t) ∈ R2. find 2 distinct solutions please11.6: Problem 8 Find the directional derivative of f(x, y, z) point (1, 2, 3) in the direction of a vector making an angle of 2 with V f(1, 2, 3). = zx + y“ at theShow that f(x, y) =(Aekx + Be-kx )(Ce2ky + De−2ky) is a solution of the wave equation∂2f/∂y2− 4∂2f/∂x2 =0.3. Verify that u(r,l) = sin(x – at) satisfies the wave equation:SEE MORE QUESTIONS