Question

In Exercises \(29-34\) define the bounded formal solution of $$ u_{x x}+u_{y y}=0, \quad 00 $$ that satisfies the given boundary conditions for general a and \(f .\) Then solve the boundary value problem for the specified a and \(f\). $$ \begin{array}{l} u_{y}(x, 0)=f(x), \quad 00 \\ a=\pi, \quad f(x)=x^{2}(\pi-x) \end{array} $$

Short answer

Based on the step-by-step solution provided, answer the following question: Question: Determine the general form of the function u(x, y) that satisfies the Poisson equation and boundary conditions. Answer: The general form of the function u(x, y) is given by: $$u(x, y) = \sum_n B_n \sin \left(\frac{n \pi}{a} x\right)e^{- \frac{n \pi}{a} y}$$ where \(B_n\) are the coefficients found by applying the boundary conditions to the eigenvalue problems for \(X(x)\) and \(Y(y)\).

Step-by-step solution

Assume Separation of Variables

Assume that the function u(x,y) can be written as a product of two functions: $$u(x, y) = X(x)Y(y)$$

Plug Separation of Variables into Poisson Equation

Substitute the product form of the solution, \(u(x,y)=X(x)Y(y)\), into the Poisson equation: $$X''(x)Y(y) + X(x)Y''(y) = 0$$

Separate the Variables

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

Constants of Integration

Since the sum of the two terms is equal to a constant, let's set each term equal to a constant: $$\frac{X''(x)}{X(x)} = -\lambda$$ $$\frac{Y''(y)}{Y(y)} = \lambda$$

Solve X(x) and Y(y) Equations

Now, solve the equations for \(X(x)\) and \(Y(y)\) separately: For \(X(x)\): $$X''(x) + \lambda X(x) = 0$$ For \(Y(y)\): $$Y''(y) - \lambda Y(y) = 0$$

Apply Boundary Conditions

Apply the boundary conditions to the equations for \(X(x)\) and \(Y(y)\) to determine the values of \(\lambda\) and the form of the solutions. The boundary conditions are: 1. \(u_y(x,0) = f(x)\) 2. \(u(0, y) = 0\) 3. \(u(a, y) = 0\) Applying the second boundary condition, we get: $$X(0)=0$$ Applying the third boundary condition, we get: $$X(a)=0$$

Eigenvalue Problem for X(x)

The equation for \(X(x)\) and the boundary conditions lead to the following eigenvalue problem: $$X''(x) + \lambda X(x) = 0, \quad X(0) = 0, \quad X(a) = 0$$ The solution to this problem can be written as: $$X_n(x) = \sin(\frac{n\pi}{a}x)$$ where \(n = 1, 2, 3, ...\) and \(\lambda_n = \left(\frac{n \pi}{a}\right)^2\)

Eigenvalue Problem for Y(y) and General Solution

The equation for \(Y(y)\) can be written as: $$Y''(y) - \lambda Y(y) = 0$$ The general solution to this can be written as the sum of all possible solutions: $$u(x, y) = \sum_n B_n \sin \left(\frac{n \pi}{a} x\right)e^{- \frac{n \pi}{a} y}$$

Apply Boundary Condition to find B_n and Solve for Specific Case

Apply the first boundary condition, \(u_y(x, 0) = f(x)\): $$f(x) = \sum_{n=1}^{\infty} -B_n \frac{n\pi}{a} \sin\left(\frac{n\pi}{a}x\right)$$ Given \(f(x) = x^2(\pi-x)\) and \(a=\pi\), we can use Fourier sine series expansion to find the coefficients \(B_n\): $$B_n = -\frac{2a}{n\pi}\int_0^a f(x)\sin \left(\frac{n\pi x}{a}\right) dx$$ $$B_n = -\frac{2\pi}{n\pi}\int_0^{\pi} x^2(\pi-x)\sin (n\pi x) dx$$ Finally, calculate the Fourier coefficients \(B_n\) and plug them back into the general solution to find the particular solution for the given values of \(a\) and \(f(x)\).

Key concepts

Partial Differential Equations

Partial Differential Equations (PDEs) are mathematical equations that involve rates of change with respect to continuous variables. Unlike ordinary differential equations, which involve derivatives with respect to a single variable, PDEs involve derivatives with respect to multiple independent variables. PDEs are essential in various fields such as physics, engineering, and finance because they model phenomena like heat conduction, sound waves, and the behavior of financial instruments.

PDEs can be linear or nonlinear, homogenous or inhomogenous. The equation in our textbook exercise, \( u_{xx} + u_{yy} = 0 \), is a homogeneous, linear PDE known as Laplace's equation, which is widely applied in electrostatics, fluid dynamics, and potential theory. Solving a PDE often involves specifying the boundary conditions, which our problem does as well. These boundary conditions are crucial as they describe how the solution behaves at the borders of the domain of interest.

Separation of Variables

Separation of Variables is a method used to solve PDEs where the solution is assumed to be the product of functions, each depending on a single coordinate direction. The idea is to reduce a PDE into multiple ordinary differential equations (ODEs) which can be solved individually.

In the textbook solution, the assumption is made that the solution \( u(x, y) \) can be represented as the product \( X(x)Y(y) \). This allows us to decouple the original PDE into simpler parts, making it more tractable. This form enables us to separate the two variables, x and y, and deal with them independently, vastly simplifying the problem.

Fourier Series

The Fourier series is a way to represent a function as a sum of sinusoidal bases. It is particularly useful in solving PDEs because many physical systems are periodic in nature. When dealing with boundary value problems that involve functions defined on an interval, the Fourier series provides a powerful tool to decompose complex periodic functions into simpler sinusoidal components.

In our example, we use the Fourier sine series to express \( f(x) = x^2(\text{π}- x) \), the function describing the boundary condition, in terms of its sinusoidal components. The coefficients \( B_n \) are calculated by integrating the product of \( f(x) \) and the sinusoidal terms, which describe the contribution of each component to the overall series. By doing so, we essentially project the function \( f(x) \) onto the basis functions, allowing for the reconstruction of the function within the boundary conditions provided.

Eigenvalue Problems

Eigenvalue problems consist of finding nontrivial solutions of equations of the form \( L\textbf{u} = \text{λ}\textbf{u} \), where \( L \) is a linear operator, \( \textbf{u} \) is an unknown vector function, and \( \text{λ} \) represents eigenvalues. In the context of our PDE exercise, the eigenvalue problem stems from applying the boundary conditions to the separated equations of \( X(x) \) and \( Y(y) \). The boundary conditions transform the problem into one that seeks solutions that satisfy both the differential equation and the constraints at the boundaries.

Particularly, the conditions \( X(0) = 0 \) and \( X(a) = 0 \) allow us to find discrete eigenvalues \( \text{λ}_n \) and corresponding eigenfunctions \( X_n(x) = \text{sin}(\frac{n\text{π}}{a}x) \). These solutions are significant since they provide the modes by which the PDE solution behaves under the given spatial constraints. The eigenfunctions form a complete set, capable of representing the range of solutions within the domain of the problem.