Advance Mathematics Numerical Analysis

a) Solve the equation ∇²u = -10(x² + y² + 10) over the square with sides x = 0, y = 0, x = 3, y = 3 with u = 0 on the boundary and mesh length 1.

b) Solve the equation ∂u/∂t = ∂²u/∂x² + ∂²u/∂y² subject to the initial condition u(x, y, 0) = sin(2πx) sin(2πy), 0 ≤ x, y ≤ 1 and the conditions u(x, y, t) = 0, t > 0 on the boundaries using. ADE method with h = 1/3 and α = 1/8 (calculate the result for one time level).

a) Solve y_xx up to t = 0.5 with a spacing of 0.1 subject to y(0, t) = 0, y(1, 0) = 0 and y(x, 0) = 10 + x(1-x).

b) Find the solution of one-dimensional wave equation.

a) Find Mellin transform of cos x.

b) Show that the Fourier sine and cosine transforms of e^-ax are G_c(ω) = √(2/π) * a / (ω² + a²), G_s(ω) = √(2/π) * ω / (ω² + a²)

a) Find the Hankel transform of r^-1e^-ar, a > 0.

b) Show that the function y(x) = (1 + x²)^-3/2 is a solution of the Volterra integral equation y(x) = (1 + x²)² / (1+x) - ∫₀^x (t / (1+x²)) y(t) dt.

a) Convert the following differential equation into integral equation: y'' + xy' + y = 0 when y(0) = y'(0) = 1.

b) Transform the boundary value problem y'' + y = x y(0) = 0, y'(1) = 0 to a Fredholm integral equation.

a) Invert the integral equation y(x) = f(x) + λ ∫₀^2π (sin x cos t) y(t) dt.

b) Solve the integral equations by the method of successive approximations: y(x) = 5x / 6 + 1/2 ∫₀^x t y(t) dt.

a) Find the general solution of the Euler equation corresponding to the functional J[y] = ∫_a^b f(x) √(1 + y'²) dx.

b) Find the extremals of the functional ∫₀¹ √(x² + y²).√(1 + y'²) dx using polar coordinates.

a) Use the Ritz method to find an approximate solution of the problem of minimizing the functional J[y] = ∫₀¹ (y'² - 2xy) dx y(0) = y(1) = 0 and compare the answer with the exact solution.