Question

In Exercises \(17-28\) define the formal solution of $$ u_{x x}+u_{u y}=0, \quad 0

Short answer

Based on the given partial differential equation (PDE) and boundary conditions, we used separation of variables to find the general solution for the PDE, and then applied the given boundary conditions to find the specific solution for the problem. For the given problem with \(a = 2\), \(b = 1\), and \(f(x) = x^2(x-2)^2\), the solution is: $$ u(x, y) = \sum_{n=1}^\infty C_n \cos \left[\frac{(n - \frac{1}{2}) \pi x}{2}\right] e^{-\left[\frac{(n - \frac{1}{2}) \pi}{2}\right]^2 y} $$ where the coefficients \(C_n\) are given by: $$ C_n = \frac{1}{2}\int_0^2 x^2(x-2)^2 \cos \left[\frac{(n - \frac{1}{2})\pi x}{2}\right]\,\mathrm{d}x $$

Step-by-step solution

Separate variables#

Assume that the solution has the form \(u(x, y) = X(x) Y(y)\). Now, we can find the derivatives and substitute them into the PDE: $$ u_{xx} = X''(x)Y(y)$$ $$ u_{uy} = X(x)Y'(y)$$ So we have: $$ X''(x)Y(y) + X(x)Y'(y) = 0$$

Separate the equation#

Divide both sides by \(X(x)Y(y)\) to separate the equation: $$ \frac{X''(x)}{X(x)} + \frac{Y'(y)}{Y(y)} = 0 $$

Set constants #

Since both sides of the equation are dependent on different variables, they must be equal to a constant. Let's set \(-\lambda^2\) as the constant: $$ \frac{X''(x)}{X(x)} = -\lambda^2 $$ $$ \frac{Y'(y)}{Y(y)} = \lambda^2 $$

Solve the ODEs #

Solve the ordinary differential equations (ODEs) for \(X(x)\) and \(Y(y)\): For \(X(x)\): $$ X''(x) + \lambda^2 X(x) = 0 $$ For \(Y(y)\): $$ Y'(y) - \lambda^2 Y(y) = 0 $$

Apply boundary conditions 3 and 4 #

Apply boundary conditions for \(u_x(0, y) = 0\) and \(u_x(a, y) = 0\): We have \(u_x(x, y) = X'(x)Y(y)\). The boundary conditions become: $$ X'(0)Y(y) = 0 \quad \text{and} \quad X'(a)Y(y) = 0 $$ Since both of these must be equal to zero, we have: $$ X'(0) = 0 \quad \text{and} \quad X'(a) = 0 $$

Determine eigenvalues and eigenfunctions for X(x) #

Solve the ODE for \(X(x)\) with the boundary conditions to get the eigenvalues and eigenfunctions: The ODE for \(X(x)\) is: $$ X''(x) + \lambda^2 X(x) = 0 $$ With boundary conditions: $$ X'(0) = 0 \quad \text{and} \quad X'(a) = 0 $$ This gives us the eigenvalues \(\lambda_n = \frac{(n - \frac{1}{2}) \pi}{a}\) and the eigenfunctions \(X_n(x) = \cos(\lambda_n x)\) for \(n = 1, 2, 3, ...\)

Solve the ODE for Y(y) #

Now we solve the ODE for \(Y(y)\): $$ Y'(y) - \lambda^2 Y(y) = 0 $$ This gives us the solution: $$ Y_n(y) = e^{-\lambda_n^2 y} $$

Construct the general solution for u(x, y) #

The general solution for \(u(x, y)\) is given by: $$ u(x, y) = \sum_{n=1}^\infty C_n \cos(\lambda_n x) e^{-\lambda_n^2 y} $$ Now we apply the remaining boundary conditions: 1. \(u(x, 0) = f(x)\) 2. \(u(x, b) = 0\)

Apply the boundary condition 1 #

Apply the boundary condition \(u(x, 0) = f(x)\): $$ f(x) = \sum_{n=1}^\infty C_n \cos(\lambda_n x) $$

Find the coefficients C_n #

Use Fourier series to find the coefficients \(C_n\): $$ C_n = \frac{2}{a}\int_0^a f(x) \cos(\lambda_n x)\,\mathrm{d}x $$ For the given \(f(x) = x^2(x-2)^2\), \(a=2\), \(b=1\), the coefficients are: $$ C_n = \frac{1}{2}\int_0^2 x^2(x-2)^2 \cos \left[\frac{(n - \frac{1}{2})\pi x}{2}\right]\,\mathrm{d}x $$

Calculate the explicit solution #

Plug the coefficients \(C_n\) back into the general solution to find the explicit solution for the given problem: $$ u(x, y) = \sum_{n=1}^\infty C_n \cos \left[\frac{(n - \frac{1}{2}) \pi x}{2}\right] e^{-\left[\frac{(n - \frac{1}{2}) \pi}{2}\right]^2 y} $$ This is the final solution for the given problem.

Key concepts

Boundary Value Problems

Boundary value problems are a crucial part of studying partial differential equations (PDEs). These problems involve finding a solution to a PDE that satisfies certain conditions at the boundaries of the domain. In this exercise, the domain is defined by the intervals \(0We have boundary conditions specifying the values of the function \(u(x,y)\) and its derivative at the edges of the domain. Specifically, \(u(x,0)=f(x)\) and \(u(x,b)=0\) describe the value of the function at the lower and upper boundaries of \(y\).
Additionally, the conditions \(u_x(0,y)=0\) and \(u_x(a,y)=0\) describe the behavior of the function's first derivative along the \(x\)-axis at the left and right boundaries.
These conditions are pivotal for determining a unique solution to the problem.

Separation of Variables

Separation of variables is a powerful method used to solve partial differential equations by reducing them to simpler, ordinary differential equations (ODEs). This technique is especially effective for linear PDEs with boundary conditions.
The essence of separation of variables is to assume that the solution can be expressed as a product of functions, each depending only on one of the independent variables. In this exercise, the solution is assumed to be of the form \(u(x, y) = X(x) Y(y)\).
This assumption allows us to substitute into the PDE and separate terms, dividing the PDE into two ODEs that depend on only one variable each.
This process facilitates finding solutions that satisfy the boundary conditions while revealing important properties like eigenvalues and eigenfunctions.

Fourier Series

Fourier series are used to express a function as a sum of sine and cosine terms, which can be particularly useful for solving PDEs with periodic or boundary value problems.
In this exercise, we employ a Fourier series to express the boundary condition \(u(x, 0) = f(x)\) in terms of cosine functions. This is because the domain and boundary conditions suggest a periodic behavior in \(x\).
This representation makes it easier to solve for the coefficients \(C_n\) in the series, which are determined by integrating \(f(x)\) multiplied by the cosine functions over the interval \([0, a]\).
Fourier series help transform complex boundary value problems into more manageable algebraic ones.

Eigenvalues and Eigenfunctions

Eigenvalues and eigenfunctions are fundamental concepts in solving differential equations. They arise naturally when applying boundary conditions to the separated equations obtained during the separation of variables.
For the differential equation related to \(X(x)\), the eigenvalues \(\lambda_n = \frac{(n - \frac{1}{2}) \pi}{a}\) are determined by the conditions \(X'(0) = 0\) and \(X'(a) = 0\).
The corresponding eigenfunctions are \(X_n(x) = \cos(\lambda_n x)\), which satisfy both the differential equation and the boundary conditions.
Eigenvalues define the possible "modes" of the solution, and eigenfunctions represent the contribution of each mode, forming a complete basis for expanding the solution in terms of Fourier series.