91Ó°ÊÓ

Approximate the solution to the wave equation $$ \begin{aligned} &\frac{\partial^{2} u}{\partial t^{2}}-\frac{\partial^{2} u}{\partial x^{2}}=0, \quad 0

Approximate the solution to the following partial differential equation using the Backward-Difference method. $$ \begin{aligned} \frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}} &=0, \quad 0

Use Algorithm \(12.1\) to approximate the solution to the elliptic partial differential equation $$ \begin{gathered} \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0, \quad 1

Approximate the solution to the following partial differential equation using the Backward-Difference method. $$ \begin{aligned} \frac{\partial u}{\partial t}-\frac{1}{16} \frac{\partial^{2} u}{\partial x^{2}} &=0, \quad 0

Approximate the solution to the wave equation $$ \begin{aligned} &\frac{\partial^{2} u}{\partial t^{2}}-\frac{\partial^{2} u}{\partial x^{2}}=0, \quad 0

Approximate the solutions to the following elliptic partial differential equations, using Algorithm 12.1: a. \(\quad \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0, \quad 0

Use the Forward-Difference method to approximate the solution to the following parabolic partial differential equations. a. $$ \begin{gathered} \frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}=0, \quad 0

Approximate the solution to the wave equation $$ \begin{aligned} &\frac{\partial^{2} u}{\partial t^{2}}-\frac{\partial^{2} u}{\partial x^{2}}=0, \quad 0

Use the Forward-Difference method to approximate the solution to the following parabolic partial differential equations. a. $$ \begin{aligned} \frac{\partial u}{\partial t}-\frac{4}{\pi^{2}} \frac{\partial^{2} u}{\partial x^{2}} &=0, \quad 0

A coaxial cable is made of a 0.1-in.-square inner conductor and a \(0.5\)-in.-square outer conductor. The potential at a point in the cross section of the cable is described by Laplace's equation. Suppose the inner conductor is kept at 0 volts and the outer conductor is kept at 110 volts. Find the potential between the two conductors by placing a grid with horizontal mesh spacing \(h=0.1\) in. and vertical mesh spacing \(k=0.1 \mathrm{in}\). on the region $$ D=\\{(x, y) \mid 0 \leq x, y \leq 0.5\\} $$ Approximate the solution to Laplace's equation at each grid point, and use the two sets of boundary conditions to derive a linear system to be solved by the Gauss-Seidel method.