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} u_{x x}+u_{y y}=0, \quad 0

Short answer

#Answer#: The solution for the given Laplace boundary value problem is: $$u(x, y) = \frac{1}{6} + \sum_{n=1}^{\infty} a_n(\cosh(n\pi x) - \text{tanh}(2n\pi)\sinh(n\pi x)) \cos(n\pi y),$$ where $$a_n = 2 \int_{0}^1 y^2(3-2y)\cos(n\pi y) dy$$

Step-by-step solution

Solve the ODE for Y(y) using boundary condition 2.

Since \(u(x, 0) = 0\), we have \(X(x)Y(0) = 0\). Since the solution can't be trivial, we set \(Y(0) = 0\). Now we have to solve the ODE for Y(y): $$Y''(y) - \kappa Y(y) = 0.$$ The given condition for \(Y(y)\) states that \(Y'(y)\) vanishes at \(y = 1\), so we substitute \(\kappa = -n^{2}\pi^{2}\) to satisfy the homogeneous Neumann boundary condition. So the general solution for \(Y(y)\) is: $$Y_n(y) = \cos(n\pi y), \quad n = 0, 1, 2, ... $$

Solve the ODE for X(x) using boundary conditions 3 and 5.

The ODE for X(x) is: $$X''(x) - n^2\pi^2 X(x) = 0, \quad n = 0, 1, 2, ... $$ Now let's use the boundary conditions 3 and 5 to solve these equations. For \(n = 0\): $$X''(x) = 0$$ The general solution for \(X(x)\) is: $$X_0(x) = Ax + B$$ For \(n > 0\): $$X''(x) = n^2\pi^2 X(x)$$ The general solution for \(X(x)\) is: $$X_n(x) = A_n\cosh(n\pi x) + B_n\sinh(n\pi x), \quad n \geq 1$$ Now we apply the boundary condition \(u_{x}(2, y) = 0\), which, since \(u(x, y) = X(x)Y(y)\), leads to the condition: $$X'(2) = 0$$ For \(n = 0\): $$A = 0 \Rightarrow X_0(x) = B$$ For \(n > 0\): $$n\pi A_n\sinh(2n\pi) + n\pi B_n\cosh(2n\pi) = 0 \\ B_n = -A_n\text{tanh}(2n\pi) \\ X_n(x) = A_n(\cosh(n\pi x) - \text{tanh}(2n\pi)\sinh(n\pi x))$$

Combine the solutions for X(x) and Y(y).

The general solution for \(u(x, y)\) is a sum of the product of \(X(x)\) and \(Y(y)\): $$u(x, y) = B + \sum_{n=1}^{\infty} A_n(\cosh(n\pi x) - \text{tanh}(2n\pi)\sinh(n\pi x)) \cos(n\pi y)$$

Find the coefficients A_n using the fourth boundary condition.

Now, let's use the boundary condition \(u(0, y) = y^2(3-2y)\): $$y^2(3-2y) = B + \sum_{n=1}^{\infty} A_n \cos(n\pi y)$$ Let's write down the Fourier cosine series for \(y^2(3-2y)\) in the interval \(0 \leq y \leq 1\): $$y^2(3-2y) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(n\pi y),$$ where $$a_n = 2 \int_{0}^1 y^2(3-2y)\cos(n\pi y) dy$$ Now we can equate the coefficients of the Fourier series to find the values of \(A_n\) and \(B\): $$B = \frac{a_0}{2}, \quad A_n = a_n$$ By integrating, we can find \(a_0\): $$a_0 = 2 \int_{0}^1 y^2(3-2y) dy = \frac{1}{3}$$ and \(a_n\): $$a_n = 2 \int_{0}^1 y^2(3-2y)\cos(n\pi y) dy$$

Write the final solution.

Now we can write the final solution for the Laplace boundary value problem: $$u(x, y) = \frac{1}{6} + \sum_{n=1}^{\infty} a_n(\cosh(n\pi x) - \text{tanh}(2n\pi)\sinh(n\pi x)) \cos(n\pi y),$$ where $$a_n = 2 \int_{0}^1 y^2(3-2y)\cos(n\pi y) dy$$

Key concepts

Laplace's Equation

Laplace's equation is a well-known second-order partial differential equation (PDE) given by \(abla^2 u = 0\), where \(u\) is the function being evaluated, and \(abla^2\) represents the Laplacian operator. In two dimensions, the equation takes the form:

• \(u_{xx} + u_{yy} = 0\)

This equation is crucial in mathematical physics and engineering, as it describes steady-state heat conduction, electrostatics, and other phenomena where steady-state implies a lack of change over time.

The main goal in solving Laplace's equation is to find a function \(u(x, y)\) that satisfies the equation throughout a given domain with specific boundary conditions.

In the exercise, the boundary conditions specify how the function behaves along the edges of the rectangular domain, which in turn influences how the solution behaves within the interior of the domain.

Fourier Series

Fourier series is a mathematical tool used to express a function as an infinite sum of sine and cosine terms. This technique is particularly useful when dealing with periodic functions or functions defined over a finite interval.

The Fourier series representation allows us to break down complex functions into simpler trigonometric functions. For a function \(f(y)\) defined over the interval \([0, 1]\), the Fourier cosine series is expressed as:

• \(f(y) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(n\pi y)\)

Fourier coefficients \(a_n\) are computed using integrals, specifically:

• \(a_n = 2 \int_{0}^{1} f(y) \cos(n\pi y) \, dy\)

In the exercise, Fourier series help to express boundary conditions mathematically so that they match the general solution \(u(x, y)\) derived from Laplace's equation.

Partial Differential Equations

Partial differential equations (PDEs) involve functions of several variables and their partial derivatives. Unlike ordinary differential equations, PDEs describe multi-variable systems like waves, heat, and fluid dynamics.

The PDE considered in the exercise is Laplace's equation, a canonical form in mathematics and physics. Solving PDEs involves finding a function that satisfies both the differential equation and the boundary conditions that accompany it.

The solution approach typically involves separation of variables—a method where functions of multiple variables are written as products of functions of single variables. Each variable part satisfies a simpler ordinary differential equation (ODE). The individual solutions of these ODEs are then combined to achieve a comprehensive solution that respects the PDE's structure and boundary conditions.

Neumann Boundary Conditions

Neumann boundary conditions specify the derivative of a function on a boundary, rather than the value of the function itself. They are crucial when you need to dictate the rate of change across a boundary.

In the context of PDEs like Laplace's equation, these conditions take the form of:

• \(\frac{\partial u}{\partial y} = 0\)

This implies no change in the \(y\)-direction at the boundary, which usually signifies an insulating or reflecting boundary in physical problems.

Neumann conditions are used in the exercise to constrain the problem further, particularly through boundaries where derivative-based properties are significant. Matching these conditions with the PDE's solutions allows us to find the specific physical behavior of a system under consideration.