Problem 1 - Applied Partial Differential Equations

Question

Consider $$ \begin{aligned} \frac{\partial u}{\partial t} &=k \frac{\partial^{2} u}{\partial x^{2}}+Q(x, t) \\ u(x, 0) &=g(x) . \end{aligned} $$ In all cases obtain formulas similar to \((8.2 .12)\) by introducing a Green's function. (a) Use Green's formula instead of term-by-term spatial differentiation if (b) Modify part (a) if \(u(0, t)=0\) and \(u(L, t)=0\). $$ u(0, t)=A(t) \quad \text { and } \quad u(L, t)=B(t) $$ Do not reduce to a problem with homogeneous boundary conditions. (c) Solve using any method if $$ \frac{\partial u}{\partial x}(0, t)=0 \quad \text { and } \quad \frac{\partial u}{\partial x}(L, t)=0 \text {. } $$ *(d) Use Green's formula instead of term-by-term differentiation if \(\frac{\partial u}{\partial x}(0, t)=A(t) \quad\) and \(\quad \frac{\partial u}{\partial x}(L, t)=B(t)\)

Step-by-Step Solution

Answer

The solution to this problem involves the use of Green's function and boundary conditions to obtain the solution of the given Partial Differential Equation. The detailed solution would require working out the math for each case given the specific boundary conditions.

Step 1: Determine the Green's Function

Firstly, it's required to determine the Green's function for this problem. Solve the associated homogeneous boundary value problem \( k G''(x, \xi) - G'(x, \xi) = 0 \) for the Green's function.

Step 2: Apply Green's Identity

Then, apply Green's Identity to the solution \( G(x, \xi) \). This will allow us to convert the problem into a simpler form, where the differential operator acts on \( G(x, \xi) \) instead of \( u(x, t) \).

Step 3: Determine Solution for u(x, t) using the Green's Function

Relationship between u(x, t) and G(x, \xi) can be written as \( u(x, t) = \int_0^L u(\xi, 0) G(x, \xi) d\xi + \int_0^T \int_0^L Q(\xi, t') G(x - \xi, t - t') d\xi dt' \).

Step 4: Apply Boundary Conditions

Substitute the boundary conditions provided into the equation obtained from the previous step. Solve to get an expression for u(x, t). Different conditions lead to different solutions.

Step 5: Repeat For each Part of the Problem

Repeat all above mentioned steps for each part of the problem (a, b, c, d) as the boundary conditions given are different for each of them.

Key Concepts

Boundary Value Problem

A boundary value problem is a mathematical issue that involves finding a function that satisfies a partial differential equation (PDE) within a certain domain while also meeting specific conditions on the domain's boundary. These conditions are known as boundary conditions, and they can significantly affect the solution to the PDE.

In the context of the exercise, the PDE involves the heat equation, which describes the distribution of heat (or temperature variation) in a given region over time. The problem is to find the temperature function \( u(x, t) \) that not only satisfies the heat equation but also meets the boundary conditions at \( x = 0 \) and \( x = L \), with \( L \) representing the length of the domain in space.

Green's Identity

Green's identity is a critical tool in solving boundary value problems involving differential equations. It relates integrals over a volume involving the Laplacian of a function and the function itself to an integral over the boundary of the volume involving the function and its normal derivative. Essentially, it converts volume integrals into surface integrals.

In our exercise, applying Green's identity allows the simplification of the problem by manipulating the equation so that derivatives act on the Green's function \( G(x, \xi) \) rather than on the unknown function \( u(x, t) \). This strategic move transforms the complex differential equation into a form that is often easier to solve.

Homogeneous Boundary Conditions

Homogeneous boundary conditions are a special type of boundary condition where the function is set to zero on the domain's boundary. In other words, they take the form \( u(x, t) = 0 \) at the boundary points. Homogeneity simplifies the problem by reducing the effect of the boundary on the solution since the solution is zero-valued at the edge.

However, in this exercise, part b explicitly states not to reduce the problem to one with homogeneous boundary conditions, underscoring that we need to tackle non-homogeneous conditions, where the value of \( u(x, t) \) at the boundary is non-zero and may depend on time \( t \).

Sturm-Liouville Problem

The Sturm-Liouville problem is a class of boundary value problems that arise in the context of differential operators that are self-adjoint. These problems play a seminal role in the theory of linear differential equations and can be expressed in the form of an eigenvalue problem involving a second-order linear differential operator. An essential feature of Sturm-Liouville problems is that they are accompanied by homogeneous boundary conditions.

Green's function, which is central to our exercise, can be used to solve Sturm-Liouville problems. Determining the Green's function typically involves solving an associated homogeneous differential equation that includes the boundary conditions.

Heat Equation

The heat equation is a PDE that describes how heat (or variations in temperature) diffuses through a given region over time. It's given by \( \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} + Q(x, t) \), where \( u(x, t) \) is the temperature, \( k \) is the thermal diffusivity constant, and \( Q(x, t) \) represents a heat source or sink within the domain.

In the given exercise, formulas similar to a reference equation are to be derived using Green's function. Green's function provides a powerful method to solve PDEs such as the heat equation, especially when the domain and boundary conditions are complex. It essentially helps to express the solution in terms of the behavior at the boundaries and properties of the heat equation's domain.