91Ó°ÊÓ

We will first solve the problem with the given boundary conditions: \(u(0, y) = 0,\ u(a, y) = f(y),\ u(x, 0) = 0,\ u(x, b) = 0.\) We assume the solution \(u(x, y) = X(x)Y(y)\), and substitute it back into Laplace's equation \((\nabla^2 u = 0)\). We will then separate the variables and find the solutions to the resulting ordinary differential equations. After setting up the equations and solving them, we obtain the solution: $$ u_1(x, y) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi x}{a}\right)\sinh\left(\frac{n\pi y}{a}\right), $$ where: $$ A_n = \frac{2}{a \sinh\left(\frac{n\pi b}{a}\right)} \int_0^a f(y) \sin\left(\frac{n\pi y}{a}\right) dy $$

Next, we will solve the problem with the given boundary conditions: \(u(0, y) = 0,\ u(a, y) = 0,\ u(x, 0) = h(x),\ u(x, b) = 0.\) We assume the solution \(u(x, y) = X(x)Y(y)\), and substitute it back into Laplace's equation \((\nabla^2 u = 0)\). We will then separate the variables and find the solutions to the resulting ordinary differential equations. After setting up the equations and solving them, we obtain the solution: $$ u_2(x, y) = \sum_{m=1}^{\infty} B_m \sin\left(\frac{m\pi x}{a}\right)\left[\cosh\left(\frac{m\pi y}{a}\right) - \frac{\cosh\left(\frac{m\pi b}{a}\right)}{\sinh\left(\frac{m\pi b}{a}\right)}\sinh\left(\frac{m\pi y}{a}\right)\right], $$ where: $$ B_m = \frac{2}{a} \int_0^a h(x) \sin\left(\frac{m\pi x}{a}\right) dx. $$

Now, we will add the solutions \(u_1(x, y)\) and \(u_2(x, y)\) to get the final solution for \(u(x, y)\): $$ u(x, y) = u_1(x, y) + u_2(x, y). $$ #Part (b): Find the solution for specific functions#

Now, we will substitute the given functions: \(h(x) = (x/a)^2\) and \(f(y) = 1 - (y/b)\), and calculate the coefficients \(A_n\) and \(B_m\) by evaluating the integrals. After evaluating the integrals, we get: $$ A_n = \frac{2b(-1)^{n+1}}{n\pi} \quad\text{and}\quad B_m = \frac{8a}{m^3 \pi^3}. $$ Now, we will substitute these coefficients into the expressions for \(u_1(x, y)\) and \(u_2(x, y)\). #Part (c): Plotting the solution for \(a = 2, b = 2\)# At this point, we have the solution for \(u(x, y)\) with the necessary coefficients. For this part, you will need to use a numerical software package (like MATLAB, Python, etc.) to plot \(u(x, y)\) for the given values of \(a = 2\) and \(b = 2\). Plot the solution as a function of \(x\), \(y\), and both \(x\) and \(y\), as well as a contour plot. As assistants can't produce graphs, create plots with 2D and 3D view as well as contour plot using a suitable programming language or software, based on the given values of \(a\) and \(b\).