Question

In Exercises \(1-16\) apply the definition developed in Example 1 to solve the boundary value problem. (Use Theorem 11.3 .5 where it applies.) Where indicated by \(\mathrm{C}\), graph the surface \(u=u(x, y), 0 \leq x \leq a\), \(0 \leq y \leq b\) $$ \begin{array}{l} \text { C } u_{x x}+u_{y y}=0, \quad 0

Short answer

Answer: The general solution to the Laplace's equation boundary value problem with given boundary conditions is: $$u(x,y) = \sum_{n=1}^{\infty} \frac{2}{n\pi^2\cosh(n\pi^2)} \cos(n\pi x)\cosh(n\pi y)$$

Step-by-step solution

Apply separation of variables

Assume that the solution \(u(x,y)\) can be written as the product of two functions, one depending on \(x\) alone and the other depending on \(y\) alone, i.e., \(u(x,y) = X(x)Y(y)\).

Plug the assumed solution into the PDE and re-arrange

Substitute the assumed solution, \(u(x,y)=X(x)Y(y)\), into the PDE \(u_{xx}+u_{yy}=0\). Then, we have \(X''(x)Y(y) + X(x)Y''(y) = 0\). Divide both sides of the equation by \(X(x)Y(y)\) to get \(\frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} = 0\).

Separate the variables

Set \(\frac{X''(x)}{X(x)} = -k^2\) and \(\frac{Y''(y)}{Y(y)} = k^2\), where \(k^2\) is a separation constant. Now we have two ordinary differential equations (ODEs) to solve: \(X''(x) = -k^2X(x)\) and \(Y''(y) = k^2Y(y)\).

Solve the ODEs and apply boundary conditions

For the first ODE, we find that the general solution can be written as \(X(x) = A\cos(kx) + B\sin(kx)\). Apply Neumann boundary condition \(u_x(0,y)=0\), and we get \(X'(0) = -Ak\sin(0) + Bk\cos(0) = Bk = 0\), which implies \(B=0\) . Therefore, \(X(x) = A\cos(kx)\). For the second ODE, the general solution can be written as \(Y(y) = C\cosh(ky) + D\sinh(ky)\). Apply Neumann boundary condition \(u_y(x,0)=0\), we get \(Y'(0)=Ck\sinh(0) + Dk\cosh(0) = Dk = 0\), which implies \(D=0\). Therefore, \(Y(y) = C\cosh(ky)\).

Formulate the general solution and apply remaining boundary conditions

So far, we have obtained \(u(x,y) = (A\cos(kx))(C\cosh(ky)) = AC\cos(kx)\cosh(ky)\), which is the solution for an arbitrary \(k\). We can now write the general solution as a sum of such solutions with different values of \(k\): $$ u(x,y) = \sum_{n=1}^{\infty} a_n \cos(n\pi x)\cosh(n\pi y)$$ Apply the mixed boundary condition \(u(x,\pi) = 0\). We get: $$\sum_{n=1}^{\infty} a_n \cos(n\pi x)\cosh(n\pi^2) = 0$$ Apply the mixed boundary condition \(u_x(1,y) = \sin y\): $$ \sum_{n=1}^{\infty} -n\pi a_n \sin(n\pi)\cosh(n\pi y) = \sin{y}$$

Find the Fourier sine series representation of the solution

In order to satisfy the last boundary condition, the right-hand side must be the Fourier sine series representation of \(\sin y\). To find the coefficients \(a_n\), first evaluate the integral \(\int_{0}^{\pi} \sin(n\pi x)\sin y dx = \frac{2}{\pi} \int_{0}^{\pi} \sin(n\pi x)\sin y dx\). The Fourier sine series coefficients are given by: $$a_n = \frac{2}{n\pi^2\cosh(n\pi^2)}$$ So, the solution to the boundary value problem is: $$u(x,y) = \sum_{n=1}^{\infty} \frac{2}{n\pi^2\cosh(n\pi^2)} \cos(n\pi x)\cosh(n\pi y)$$

Key concepts

partial differential equations

Partial differential equations (PDEs) are a type of mathematical equation that involve multiple independent variables. In a PDE, an unknown function depends on several independent variables and the equation involves partial derivatives with respect to these variables. A classic form of PDEs is written as:

• \(u_{xx} + u_{yy} = 0\)

This equation shows that it involves second derivatives of the unknown function \(u(x, y)\) with respect to \(x\) and \(y\). Such equations often model physical phenomena, like heat or wave propagation, where you need to understand how a field like temperature or displacement varies in space at given times. In the example problem, we are solving a PDE with specific boundary conditions, which specify the behavior of the solution on the boundary of the problem's domain.

Fourier series

The Fourier series is a way to represent a function as the sum of sines and cosines. This technique is useful in solving PDEs by breaking down complex oscillations into simpler components. In the boundary problem given, after separation of variables, we find expressions that involve sums of cosine and hyperbolic cosine functions, which are solved using what would essentially become a Fourier series.

• The general solution \(u(x,y) = \sum_{n=1}^{\infty} a_n \cos(n\pi x)\cosh(n\pi y)\) is an example of a Fourier series representation.

This decomposition simplifies the solution process, allowing us to solve the PDE by examining the behavior of its components, each corresponding to a different frequency, indicated by \(n\). Fourier series are particularly useful because they transform the boundary conditions into algebraic equations for the coefficients \(a_n\), which simplify the problem considerably.

separation of variables

Separation of variables is a mathematical method for solving partial differential equations, where a complex PDE is broken into simpler ordinary differential equations (ODEs). The method begins by assuming that the solution is the product of two or more single-variable functions. For the given boundary problem:

• Assume \(u(x, y) = X(x)Y(y)\), separate the variables after substituting into the equation, resulting in two ODEs: \(X''(x) = -k^2X(x)\) and \(Y''(y) = k^2Y(y)\).

Separation of variables works well for linear, homogeneous PDEs with constant coefficients and well-defined boundary conditions. This method essentially reduces a multidimensional problem to one that involves functions of a single variable, making the problem more manageable and allowing for solutions using standard techniques for solving ODEs.

ordinary differential equations

Ordinary differential equations (ODEs) are equations involving a function and its derivatives but are related to only one independent variable. In the context of our PDE problem, after applying separation of variables, we end up with two types of ODEs:

• \(X''(x) = -k^2X(x)\)
• \(Y''(y) = k^2Y(y)\)

Each of these equations is dependent on a single variable, either \(x\) or \(y\), and involves ordinary derivatives. Solving these ODEs requires using techniques for characteristic equations, which might result in sine, cosine, exponential, or hyperbolic solutions, depending on the nature of the ODE. These solutions then satisfy the boundary conditions of the PDE, which are crucial for determining specific values or forms of the solution. Incorporating the results back into the assumed form \(u(x,y) = X(x)Y(y)\) allows for the construction of a full solution to the original boundary value problem.