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 \(f\). $$ \begin{array}{l} u_{y}(x, 0)=f(x), \quad 00 \\ a=7, \quad f(x)=x(7-x) \end{array} $$

Short answer

#tag_title#Short Answer#tag_content#The specific solution to the given boundary value problem can be represented as: $$u(x, y) = \sum_{n=1}^{\infty}A_n\cos\left(\frac{n\pi x}{7}\right)e^{-\frac{n\pi y}{7}}$$ where the coefficients \(A_n\) are given by: $$A_n = \frac{2}{7}\int_0^7 x(7-x)\cos\left(\frac{n\pi x}{7}\right)dx$$ To find the complete solution for this problem, one must evaluate the integral for \(A_n\) and apply the given values for \(a = 7\) and \(f(x) = x(7-x)\).

Step-by-step solution

Rewrite the PDE in terms of Separation of Variables

Since the Laplace operator \((u_{xx} + u_{yy})\) is linear and the boundary is rectangular, we can try to apply the method of separation of variables. Let \(u(x, y) = X(x)Y(y)\). Substituting this expression into the Laplace equation, we get: $$X''(x)Y(y) + X(x)Y''(y) = 0$$

Divide by the product of the functions

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

Introduce a separation constant

Since the left-hand side of the equation depends only on \(x\), and the right-hand side depends only on \(y\), introduce a separation constant \(-k^2\) such that: $$\frac{X''(x)}{X(x)} = -k^2$$ and $$\frac{Y''(y)}{Y(y)} = k^2$$

Solve the ODEs for X and Y

To solve each of these Ordinary Differential Equations (ODEs) for \(X(x)\) and \(Y(y)\), we first solve the ODE for \(X(x)\): $$X''(x) = -k^2X(x)$$ The general solution to this ODE is given by \(X(x) = A\cos(kx) + B\sin(kx)\). Next, we solve the ODE for \(Y(y)\): $$Y''(y) = k^2Y(y)$$ The general solution to this ODE is given by \(Y(y) = Ce^{ky} + De^{-ky}\).

Apply the boundary conditions

Applying the given boundary conditions for the \(x\) variable: \(u_x(0,y) = X'(0)Y(y) = 0 \Rightarrow X'(0) = 0\) This implies \(B = 0\), so \(X(x) = A\cos(kx)\). For the \(y\) variable, as \(y\to \infty\), we want our solution to be bounded. Therefore, we require \(Ce^{ky} = 0\). So, the solution for \(Y(y)\) becomes \(Y(y) = De^{-ky}\).

Construct the general solution

The general formal solution for the problem is now given by \(u(x, y) = X(x)Y(y) = (A\cos(kx))(De^{-ky})\). To find the specific solution for this problem, we need to use the given values for \(a = 7\) and \(f(x) = x(7-x)\).

Consider Fourier sine series

The boundary condition, \(u_y(x, 0) = f(x)\), can be represented in terms of the Fourier sine series of the function \(f(x)\). We can write \(u(x, y) = \sum_{n=1}^{\infty}A_n\cos\left(\frac{n\pi x}{a}\right)e^{-\frac{n\pi y}{a}}\). The boundary condition gives us: $$\sum_{n=1}^{\infty}A_n\left(-\frac{n\pi}{a}\right)e^{-\frac{n\pi y}{a}}\cos\left(\frac{n\pi x}{a}\right) = f(x)$$

Obtain the specific solution for A_n

To find the coefficients \(A_n\) for the specific solution, apply the given values for \(a\) and \(f(x)\): $$a=7, \quad f(x)=x(7-x)$$ Derive the Fourier cosine series by integrating over the given interval \((0,a)\) with the orthogonal functions, i.e., $$A_n = \frac{2}{a}\int_0^a f(x)\cos\left(\frac{n\pi x}{a}\right)dx$$ For this specific problem, we have \(A_n = \frac{2}{7}\int_0^7 x(7-x)\cos\left(\frac{n\pi x}{7}\right)dx\). Find \(A_n\) by evaluating the integral and use it to construct the specific solution \(u(x, y)\) for the given boundary value problem.

Key concepts

Boundary Value Problems

Boundary Value Problems involve finding a solution to a differential equation that satisfies certain specified conditions, which are known as boundary conditions. These conditions are specified at the boundaries of the domain of the problem.

In our exercise, we're dealing with the Laplace equation, a common partial differential equation (PDE), in a rectangular domain. The boundary condition for this problem includes:

• On the bottom boundary, we have a condition relating to the derivative with respect to y: \(u_{y}(x, 0)=f(x)\).
• On the sides, the condition is \(u_{x}(0, y)=0\) and \(u(a, y)=0\).

These conditions ensure that the solution is unique and satisfies the physical or geometrical constraints of the problem.

The specified boundary conditions are crucial because they allow us to find the particular solution of the PDE with meaningful physical significance.

Separation of Variables

Separation of Variables is a method for solving partial differential equations, where a multivariable equation is separated into simpler, single-variable problems.

In this approach, we express the solution as a product of functions, each depending on a single coordinate, e.g., \(u(x, y) = X(x)Y(y)\).

When substituting this product into the Laplace equation \(u_{xx} + u_{yy} = 0\), it separates into equations that will only involve a single coordinate:

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

This method significantly simplifies the problem by reducing PDEs into two ordinary differential equations (ODEs), each solvable via well-known techniques.

Separation of Variables is particularly powerful in situations with simple geometric boundaries, like rectangles or circles, where each boundary condition can be independently applied.

Laplace Equation

The Laplace Equation, a second-order partial differential equation, is expressed as \( abla^2 u = 0 \) or \( u_{xx} + u_{yy} = 0 \) in two dimensions.

It describes a wide variety of steady-state phenomena like heat distribution, electrostatics, and fluid flow, making it a fundamental equation in both mathematics and physics.

In our case, the Laplace equation in a bounded domain needs a solution that satisfies the specified boundary conditions. By using separation of variables, we transform this equation into solvable forms. Specifically, it becomes a combination of trigonometric functions in the x-coordinate and exponential functions in the y-coordinate.

The power of the Laplace equation lies in its ability to model equilibrium states, often requiring sophisticated techniques to tackle the unique boundary conditions of practical problems.

Fourier Series

Fourier Series is a method for representing a function as a sum of periodic components, and it is very handy in solving boundary value problems.

In contexts like our exercise, the Fourier Series comes into play when we're expressing the function \(f(x)\) at the boundary as a sum of sines or cosines.

For a specific boundary condition, such as \(u_y(x, 0) = f(x)\), Fourier sine series is used. It approximates \(f(x)\) as an infinite sum of sine functions. These sine functions are determined by coefficients calculated through integrals over the function \(f(x)\). These coefficients are known as \(A_n\), obtained by:

• \(A_n = \frac{2}{a}\int_0^a f(x)\cos\left(\frac{n\pi x}{a}\right)dx\)

This expression helps construct a form of the particular solution for the boundary problem. Fourier Series enable us to represent complex, periodic behaviors and solve problems in bounded domains effectively.