Question

In Exercises \(17-28\) define the formal solution of $$ u_{x x}+u_{u y}=0, \quad 0

Short answer

Question: Find the formal solution to the boundary value problem with the given PDE and boundary conditions: PDE: \(u_{xx} + u_{uy} = 0\) Boundary conditions: \(u(x, 0) = x(6-x)\), \(u(x, 2) = 0\), \(u(0, y) = 0\), and \(u_x(3, y) = 0\). Answer: The formal solution to the boundary value problem is: $$u(x, y) = \sum_{n=1}^\infty \frac{12}{n^2\pi^2}\left((-1)^n - 3^n(-1)^n -1 \right) \cdot \sin\left({n \pi x \over 3}\right) \cdot e^{\left(\frac{n\pi}{3}\right)^2 y}$$

Step-by-step solution

Identify the PDE and boundary conditions

The givens are: PDE: \(u_{xx} + u_{uy} = 0\) Boundary conditions: \(u(x, 0) = f(x)\), \(u(x, b) = 0\), \(u(0, y) = 0\), and \(u_x(a, y) = 0\).

Separate the variables

We rewrite the PDE as: \({u_{xx} \over u} + {u_{uy} \over u} = 0\). Now, let \(u(x, y) = X(x)Y(y)\). Then, the separated equation will be: \({X''(x) \over X(x)} + {Y'(y) \over Y(y)} = 0\).

Transpose the separated equation

We now rewrite the separated equation as two ordinary differential equations (ODEs): $${X''(x) \over X(x)} = -\lambda^2$$ $${Y'(y) \over Y(y)} = \lambda^2$$

Solve the ODEs

Solving the ODEs we get: 1. The equation for \(X(x)\): $$X''(x) + \lambda^2 X(x) = 0$$ General solution: \(X(x) = A\cos(\lambda x) + B\sin(\lambda x)\) 2. The equation for \(Y(y)\): $$Y'(y) - \lambda^2 Y(y) = 0$$ General solution: \(Y(y) = Ce^{\lambda^2 y}\)

Apply boundary conditions

Now, we apply boundary conditions and solve for \(A, B, C, \lambda\): 1. \(u(0, y) = 0 \implies X(0)Y(y) = 0 \implies X(0)=0\), so \(A=0\) and $$X(x) = B\sin(\lambda x)$$ 2. \(u_x(a, y) = 0 \implies X'(a) Y(y) = 0\) $$\lambda B \cos(\lambda a) = 0 \implies \lambda a = \pi \implies \lambda = {\pi \over a}$$

Form the general solution

Using the values of \(A, B, \lambda\), we can now write the general solution to the PDE as: $$u(x, y) = \sum_{n=1}^\infty B_n \sin\left({n \pi \over a} x\right)e^{\left(\frac{n\pi}{a}\right)^2 y}$$

Apply remaining boundary conditions

Using the remaining boundary conditions: 1. \(u(x, 0) = f(x) \implies \sum_{n=1}^\infty B_n \sin\left({n \pi \over a} x\right) = f(x)\) 2. \(u(x, b) = 0 \implies \sum_{n=1}^\infty B_n \sin\left({n \pi \over a} x\right)e^{\left(\frac{n\pi}{a}\right)^2 b} = 0\)

Solve for specific values

Now let's solve the boundary value problem for the specified values: $$a=3, \quad b=2, \quad f(x)=x(6-x)$$ Us the Fourier Sine series for finding the coefficients: $$B_n = \frac{2}{a} \int_0^a x(6-x) \sin\left({n \pi x \over a}\right) dx = {12\over n^2\pi^2}\left((-1)^n - 3^n(-1)^n -1 \right)$$

Final solution

The final solution to the boundary value problem is: $$u(x, y) = \sum_{n=1}^\infty \frac{12}{n^2\pi^2}\left((-1)^n - 3^n(-1)^n -1 \right) \cdot \sin\left({n \pi x \over 3}\right) \cdot e^{\left(\frac{n\pi}{3}\right)^2 y}$$

Key concepts

Partial Differential Equations

Partial differential equations (PDEs) involve functions with more than one independent variable and contain partial derivatives. In the context of the exercise, the PDE is given as \( u_{xx} + u_{uy} = 0 \). This describes the relationship between a function \( u(x, y) \) and its partial derivatives with respect to \( x \) and \( y \).
PDEs are crucial in modeling multidimensional systems such as fluid flow, heat transfer, and wave propagation. They describe the behavior of physical systems under various conditions.
To solve PDEs, we often require boundary conditions, which give additional information needed to determine a unique solution. For this exercise, boundary conditions like \( u(x, 0) = f(x) \) and \( u(x, b) = 0 \) help in solving for the function \( u(x, y) \).

Fourier Series

Fourier series is a mathematical tool to represent a periodic function as a sum of sine and cosine terms. It decomposes complex functions into a series of simpler sine and cosine functions.
In boundary value problems, Fourier series are especially useful for solving linear PDEs with boundary conditions. In our problem, the solution involves the Fourier sine series, given the separation of variables results in functions that are sinusoidal in nature.
The Fourier coefficients are determined by integrating the product of the function and sine terms over a specific interval. Here, the coefficients \( B_n \) are derived from \[ B_n = \frac{2}{a} \int_0^a x(6-x) \sin\left(\frac{n \pi x}{a}\right) dx \]
The series converges to the function \( f(x) \) within a domain specified by the boundary conditions.

Separation of Variables

The technique of separating variables benefits solving many PDEs by reducing them to a set of simpler ordinary differential equations (ODEs). In this exercise, the solution assumes \( u(x, y) = X(x)Y(y) \). This implies that the function will be expressed as a product of functions, each depending on a single variable.
By substituting this form into the original PDE, we separate the equation into parts that only involve \( x \) and those that only involve \( y \). This leads to \[ \frac{X''(x)}{X(x)} + \frac{Y'(y)}{Y(y)} = 0 \]
The solution involves finding \( X(x) \) and \( Y(y) \) from the separate ODEs, which helps greatly in satisfying specific boundary conditions. The separated equations are simpler and more manageable to solve, providing a direct pathway to reach the general and specific solutions of \( u(x, y) \).

Mathematical Modeling

Mathematical modeling involves using mathematical equations and structures to represent real-world phenomena. In this exercise, we build a model described by a boundary value problem, reflecting how variables \( x \) and \( y \) interact under specific conditions.
This type of modeling helps translate physical scenarios—such as temperature distribution, vibration of strings, or electrostatics fields—into mathematical terms, allowing us to use analytical methods for prediction and analysis.
By defining conditions like \( u(x, 0) = f(x) \) and \( u(x, b) = 0 \), we specify how our model should behave at certain points, yielding a solution that fits the modeled situation reliably. Solving the model often involves PDEs and methods like separation of variables and Fourier series to ensure accurate depiction and computational efficiency.
Overall, mathematical modeling is an essential component in translating theoretical constructs to practical, applicable solutions.