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 $u=u(x,y),0\le x\le a$, $0\le y\le b$ $$ \begin{array}{ll} u_{x x}+u_{y y}=0, \quad 0

Short answer

Question: Determine the solution of the following boundary value problem: Laplace's equation is given by: $$u_{xx}+u_{yy}=0,$$ with $0<x<3$ and $0<y<2$. Boundary conditions are: - Neumann boundary condition: $u_y(x,2)=x^2$, for $0\le x\le 3$ - Dirichlet boundary condition: $u(x,0)=0$, for $0<x<3$ - Neumann boundary condition: $u_x(0,y)=0$, for $0<y<2$ - Neumann boundary condition: $u_x(3,y)=0$, for $0<y<2$ Answer: The solution of the boundary value problem is given by: $$u(x,y)=\sum _{n=1}^{\mathrm{\infty }}\frac{9(-1{)}^{n+1}}{n\pi }\frac{(3-n\pi )(3+n\pi )}{4\cosh \left(\frac{2n\pi }{3}\right)}\cos \left(\frac{n\pi x}{3}\right)\sinh \left(\frac{n\pi (2-y)}{3}\right).$$

Step-by-step solution

Identify the PDE and domain

We are given Laplace's equation: $$u_{xx}+u_{yy}=0,$$ with $0<x<3$ and $0<y<2$.

Apply separation of variables

We will try a solution of the form $u(x,y)=X(x)Y(y)$. Plugging this into Laplace's equation and dividing by $u(x,y)$, we get: $$\frac{X^{″}(x)}{X(x)}+\frac{Y^{″}(y)}{Y(y)}=0.$$ Since the left side is a function of $x$, and the right side is a function of $y$, they must be equal to a constant. Thus, we have two ODEs: $$X^{″}(x)=-\lambda X(x),Y^{″}(y)=(\lambda -0)Y(y),$$ where $\lambda$ is an eigenvalue.

Set up the eigenvalue problem for X(x)

The boundary conditions for $X(x)$ are: $$X^{\prime }(0)=0,X^{\prime }(3)=0.$$ The ODE for X(x) can be written as: $$X^{″}(x)+\lambda X(x)=0.$$ Solving this eigenvalue problem will give us the eigenvalues $\lambda$ and eigenfunctions $X_n(x)$.

Solve the eigenvalue problem for the eigenfunctions X(x)

The eigenvalue problem has solutions in the form: $$X_n(x)=\cos \left(\frac{n\pi x}{3}\right),$$ where $n=1,2,3,\dots$. These eigenfunctions form an orthogonal basis on the domain $0<x<3$.

Solve the ODE for Y(y) with these eigenvalues

Now we solve the equation: $$Y^{″}(y)+{\left(\frac{n\pi }{3}\right)}^{2}Y(y)=0,0<y<2,$$ with the boundary condition $Y(0)=0$. The general solution of this ODE is: $$Y_n(y)=\sinh \left(\frac{n\pi (2-y)}{3}\right),$$ which satisfies the boundary condition at $y=0$.

Formulate the general solution

As per separation of variables, the general solution is given by the sum of products of these eigenfunctions: $$u(x,y)=\sum _{n=1}^{\mathrm{\infty }}{C}_n\cos \left(\frac{n\pi x}{3}\right)\sinh \left(\frac{n\pi (2-y)}{3}\right).$$

Determine the coefficients $C_n$

We have the Neumann boundary condition: $$u_y(x,2)=x^2,0\le x\le 3.$$ Differentiating the general solution with respect to y and substituting $y=2$, we obtain: $$\sum _{n=1}^{\mathrm{\infty }}-C_n\frac{n\pi }{3}\cos \left(\frac{n\pi x}{3}\right)\cosh \left(\frac{2n\pi }{3}\right)=x^2.$$ Since the cosine functions form an orthogonal basis, we can find the coefficients by multiplying both sides by $\cos \left(\frac{m\pi x}{3}\right)$ and integrating from $0$ to $3$, and using the orthogonality property of cosine functions. We get: $$C_m=\frac{9(-1{)}^{m+1}}{m\pi }\frac{(3-m\pi )(3+m\pi )}{4\cosh \left(\frac{2m\pi }{3}\right)}.$$

Write the final solution

Now we can plug the coefficients back into our general solution to obtain the final answer: $$u(x,y)=\sum _{n=1}^{\mathrm{\infty }}\frac{9(-1{)}^{n+1}}{n\pi }\frac{(3-n\pi )(3+n\pi )}{4\cosh \left(\frac{2n\pi }{3}\right)}\cos \left(\frac{n\pi x}{3}\right)\sinh \left(\frac{n\pi (2-y)}{3}\right).$$

Key concepts

Partial Differential Equations

Partial Differential Equations (PDEs) are mathematical equations that involve functions of multiple variables and their partial derivatives. They are used to formulate problems involving functions of several variables and are either solved by analytical or numerical methods. PDEs are key in describing various phenomena in fields such as physics, engineering, and finance. Examples include the heat equation, wave equation, and Schrödinger equation. These equations describe how physical quantities such as temperature, pressure, or wave functions evolve over time or space.

One common approach to solving PDEs is by using separation of variables, which is particularly useful for linear PDEs with certain boundary conditions. This method assumes that the solution can be expressed as a product of functions, each of a single coordinate. This technique transforms a complex PDE into simpler, ordinary differential equations (ODEs), which are easier to solve. Overall, PDEs provide a powerful tool for modeling real-world systems with multiple interdependent variables and dynamic changes.

Laplace's Equation

Laplace's Equation is a specific type of partial differential equation that appears in various fields such as electromagnetism, fluid dynamics, and potential theory. It is expressed as: $$u_{xx}+u_{yy}=0$$This equation characterizes the behavior of scalar potentials, such as electric potentials and gravitational potentials, in a steady-state scenario, meaning it describes systems in equilibrium without time dependence.

In the context of boundary value problems, Laplace's Equation is solved within a specific domain subject to boundary conditions, which define the values of the potential along the border of the domain. Common boundary conditions include Dirichlet conditions, which fix the function value, and Neumann conditions, which fix the derivative (flux) at the boundary.

Laplace's Equation is significant in its simplicity and universality, providing insight into more complex problems by addressing the core steady-state behaviors.

Separation of Variables

Separation of Variables is a method used to solve partial differential equations, allowing an equation to be broken down into simpler, separate problems. Typically, this technique is applied when the equation and boundary conditions exhibit a level of symmetry or homogeneity. The main idea is to assume that the solution can be represented as a product of functions, each depending solely on one of the independent variables.

For instance, in solving Laplace’s Equation, you can consider a solution of the form $u(x,y)=X(x)Y(y)$. This assumption results in transforming the original PDE into two separate ordinary differential equations (ODEs). Each of these ODEs then corresponds to one of the variables. This reduction makes finding solutions more manageable, as ODEs are generally simpler to resolve through standard techniques.

The ultimate goal is to satisfy both the PDE and any boundary conditions specified. The solution often involves combining the outcomes of several eigenvalue problems, each contributing to a part of the solution. This method is particularly practical for problems with straightforward geometries like rectangles or cylinders.

Eigenvalue Problems

Eigenvalue problems are critical components in the solution process of partial differential equations, especially when using the method of separation of variables. They often arise when conditions call for solutions that oscillate or decay, as seen in problems involving waves, heat flow, and quantum mechanics.

In mathematical terms, an eigenvalue problem involves finding solutions for equations of the general form:$$A\mathbf{v}=\lambda \mathbf{v}$$where $A$ is a linear operator, $\mathbf{v}$ is the eigenvector, and $\lambda$ is the eigenvalue.

For PDEs, the solution process usually involves transforming the spatial component into a problem involving a differential operator. The boundary conditions then determine which eigenvalues are admissible and influence the corresponding eigenfunctions. Solving the eigenvalue problem gives a set of eigenvalues and eigenfunctions, which often form a complete basis set that can be used to express the general solution to the PDE.

These solutions are crucial as they describe modes of variations within the system being studied, allowing for decomposition into simpler, understandable components.