Differential Equations - II & Laplace Transform Study Notes for Civil Engineering

Initial & Boundary Value Problems

With initial value problems, we had a differential equation and we specified the value of the solution and an appropriate number of derivatives at the same point (collectively called initial conditions).
For instance, for a second order differential equation, the initial conditions are,

With boundary value problems we will have a differential equation and we will specify the function and/or derivatives at different points, which we’ll call boundary values. For second order differential equations, which will be looking at pretty much exclusively here, any of the following can, and will, be used for boundary conditions.

- eq2

eq3

eq4

So, for the purposes of our discussion here we’ll be looking almost exclusively at differential equations in the form,

eq-5

In the earlier chapters, we said that a differential equation was homogeneous if for all x.
Here we will say that a boundary value problem is homogeneous if in addition to we also have and (regardless of the boundary conditions we use).
If any of these are not zero we will call the BVP nonhomogeneous.
It is important to now remember that when we say homogeneous (or nonhomogeneous) we are saying something not only about the differential equation itself but also about the boundary conditions as well.

Example: Solve the following BVP.

Solution:

Okay, this is a simple differential equation so solve and so we’ll leave it to you to verify that the general solution to this is,

Now all that we need to do is apply the boundary conditions.

The solution is then,

Example 2 : Solve the following BVP.

Solution:

We’re working with the same differential equation as the first example so we still have,

Upon applying the boundary conditions we get,

So in this case, unlike previous example, both boundary conditions tell us that we have to have and neither one of them tell us anything about .
Remember however that all we’re asking for is a solution to the differential equation that satisfies the two given boundary conditions and the following function will do that,

In other words, regardless of the value of we get a solution and so, in this case we get infinitely many solutions to the boundary value problem.

Laplace Transform

The Laplace transform , for all s such that this integral converges.

Examples: , . .

Table of Laplace transform of function:

Theorem (Linearity of the Laplace transform):
- Suppose and are defined for , and and are real numbers.
- Then for .
Inverse Laplace transform
- Given a function G, a function g such that is called an inverse Laplace transform of G.
- In this event, we write .
Lerch Theorem
- Let f and g be continuous on and suppose that . Then .
Theorem: If and and and are real numbers, then .

Solutions of IVP using Laplace Transform

Let f be continuous on and suppose is piecewise continuous on for every positive k.
Suppose also that if . Then .
Theorem (Laplace transform of a higher derivative)
- Suppose are continuous on and is piecewise continuous on for every positive k.
- Suppose also that for and for . Then .

Examples: .

Convolution

If f and g are defined on , then the convolution of f with g is the function defined by for .
Convolution theorem): If is defined, then .
Theorem: Let and . Then .

Example: .

Determine f such that .

Theorem: If is defined, so is , and .
Example:

Solve .

Unit Impulses & DIRAC's Delta Function

Dirac’s delta function

where

Filtering property
Let and let f be integrable on and continuous at a. Then
Let the definition of the Laplace transformation of the delta function.

Laplace Transform Solution of Systems

Example: Solve the system

Differential Equations with Polynomial Coefficients

Let for and suppose that F is differentiable. Then for .
Corollary: Let for and let n be a positive integer. Suppose F is n times differentiable. Then for .

Example

Let f be piecewise continuous on for every positive number k and suppose there are numbers M and b such that for . Let . Then .

The Wave Equation

One dimensional Wave Equation

The variable t has the significance of time, the variable x is the spatial variable. Unknown function u(x,t) depends both of x and t.
For example, in case of vibrating string the function u(x,t) means the string deviation from equilibrium in the point x at the moment t.
Consider the wave equation for the vibrating string in more details.
Applying the 2d Newton’s law to the portion of the string between points x and we get the equation

and dividing by the and taking the limit we obtain the wave equation

If both ends of the string are fixed then the boundary conditions are

and the initial conditions are

In case of infinite string the wave equation with initial conditions

has solution (D’Alembert’s solution):

Diffusion or Heat Equation

The equation for this distribution is

Now we consider the temperature distribution, which depends on the point x of the rod and time t.

In case of diffusion equation r(x)=0.
Initial temperature distribution:
If the function f(x) is so-called delta function (at the initial moment we have the heat source at one point with coordinate ) then the homogeneous (r(x)=0) heat equation

has the solution.

This is the so-called fundamental solution.

Example: The Braselton (chemical reaction system with two components) on the unit interval

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Next Subject - Integral Calculus

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