Problem 14 - Applied Partial Differential Equations

Question

Consider \(L(u)=f(x)\) with \(L=\frac{d}{d x}\left(p \frac{d}{d x}\right)+q\). Assume that the appropriate Green's function exists. Determine the representation of \(u(x)\) in terms of the Green's function if the boundary conditions are nonhomogeneous: (a) \(u(0)=\alpha \quad\) and \(\quad u(L)=\beta\) (b) \(\frac{d u}{d x}(0)=\alpha \quad\) and \(\quad \frac{d u}{d x}(L)=\beta\) (c) \(u(0)=\alpha \quad\) and \(\quad \frac{d u}{d x}(L)=\beta\) *(d) \(u(0)=\alpha \quad\) and \(\quad \frac{d u}{d x}(L)+h u(L)=\beta\)

Step-by-Step Solution

Answer

The solutions are: (a) \(u(x) = \int_{0}^{L} G(x,t) f(t) dt + \alpha G(x, 0) + \beta G(x, L)\) (b) \(u(x) = \int_0^L G(x,t) f(t) dt - \alpha \frac{dG}{dx}(x, 0) - \beta \frac{dG}{dx}(x, L)\) (c) \(u(x) = \int_0^L G(x,t) f(t) dt + \alpha G(x, 0) - \beta \frac{dG}{dx}(x, L)\) (d) \(u(x) = \int_0^L G(x,t) f(t) dt + \alpha G(x, 0) - (\beta - h\alpha) \frac{dG}{dx}(x, L)\)

Step 1: Define the general expression

Let's consider a general linear differential operator \(L\) with boundary conditions \(a_1\) and \(a_2\). The problem \(Ly = f(x)\) with boundary conditions \(a_1\) and \(a_2\) is equivalent to saying that \(y(x) = \int_{a_1}^{a_2} GR(x,t) \cdot f(t) dt\), where \(GR(x,t)\) is the Green's function.

Step 2: Apply to each case

Given the differential operator \(L\) in our exercise and knowing that the boundaries are \(a_1 = 0\) and \(a_2 = L\), we can write: (a) For \(u(0) = \alpha\) and \(u(L) = \beta\) , \(u(x) = \int_{0}^{L} G(x,t) f(t) dt + \alpha G(x, 0) + \beta G(x, L)\) (b) For \(\frac{du}{dx}(0) = \alpha\) and \(\frac{du}{dx}(L) = \beta\) , \(u(x) = \int_0^L G(x,t) f(t) dt - \alpha \frac{dG}{dx}(x, 0) - \beta \frac{dG}{dx}(x, L)\) (c) For \(u(0) = \alpha\) and \(\frac{du}{dx}(L) = \beta\) , \(u(x) = \int_0^L G(x,t) f(t) dt + \alpha G(x, 0) - \beta \frac{dG}{dx}(x, L)\) (d) For \(u(0) = \alpha\) and \(\frac{du}{dx}(L) + hu(L) = \beta\) , \(u(x) = \int_0^L G(x,t) f(t) dt + \alpha G(x, 0) - (\beta - h\alpha) \frac{dG}{dx}(x, L)\)

Key Concepts

Partial Differential Equations

Partial Differential Equations (PDEs) are a type of mathematical equation that involve rates of change with respect to more than one independent variable. They are used extensively to describe various physical phenomena such as heat conduction, wave propagation, fluid dynamics, and quantum mechanics.

PDEs can be challenging to solve, as they often require complex boundary and initial conditions. In the context of our exercise, the differential operator, represented by a second-order PDE'
\
L(u) = f(x) \
\
with
\
L = \frac{d}{dx}\left(p \frac{d}{dx}\right) + q \
\
is used to model a system where \(u(x)\) is the unknown function we are seeking, and \(f(x)\) is a source term. The Green's function method provides a powerful tool to solve such PDEs, especially when dealing with nonhomogeneous boundary conditions, as highlighted in our exercise.

Boundary Value Problems

Boundary Value Problems (BVPs) arise when we are looking for a solution to a differential equation subject to specific conditions at the boundaries of the domain. These problems are a subset of PDEs where the solution must satisfy not just the equation, but also adhere to these pre-set conditions.

In the given exercise, the boundary conditions vary across each scenario:

• For (a) and (c), we have Dirichlet conditions, which specify the value of the solution at the boundary.
• For (b), we have Neumann conditions, which specify the value of the derivative of the solution at the boundary.
• For (d), we have a Robin condition, which is a weighted combination of Dirichlet and Neumann conditions.

Each type of boundary condition requires a different approach when utilizing the Green's function to represent the solution, as it affects the form of the function and the final integral expression for \(u(x)\).

Differential Operators

Differential operators are symbols that represent the operation of differentiation in mathematical expressions. They play a crucial role in formulating both ordinary and partial differential equations and can vary in complexity.

In our exercise, the differential operator is given by:
\
L = \frac{d}{dx}\left(p \frac{d}{dx}\right) + q \
\
This operator involves the first derivative of a product of the function \(p(x)\) and the second derivative of our unknown function \(u(x)\), as well as a potential term \(q(x)\). Such operators are not just mathematical abstractions but also represent physical processes. For instance, \(p(x)\) could represent material properties that vary with position, and \(q(x)\) could be a potential field or source term within the system being modeled.

The proper use and understanding of differential operators are crucial for setting up and solving PDEs, as they describe the nature of the spatial variations of our unknown function within the domain of interest.