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.

             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. 

When solving linear initial value problems a unique solution will be guaranteed under very mild conditions. With boundary value problems we will often have no solution or infinitely many solutions even for very nice differential equations that would yield a unique solution if we had initial conditions instead of boundary conditions.

So, with some of the basic stuff out of the way let’s find some solutions to a few boundary value problems. Note as well that there really isn’t anything new here yet.  We know how to solve the differential equation and we know how to find the constants by applying the conditions.  The only difference is that here we’ll be applying boundary conditions instead of initial conditions.

We mentioned above that some boundary value problems can have no solutions or infinite solutions we had better do a couple of examples of those as well here.  This next set of examples will also show just how small of a change to the BVP it takes to move into these other possibilities.

2.1       Definition and Basic Properties:

Initial value problem  algebra problem  solution of the algebra problem  solution of the initial value                                                 problem

1. Definition (Laplace Transform): The Laplace transform , for all s such that this integral converges.

Examples:   , .    .

2. Table of Laplace transform of functions

3. Theorem (Linearity of the Laplace transform): Suppose  and  are defined for , and  and  are real numbers.  Then  for .

4. Definition (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 .

5. Theorem (Lerch): Let f and g be continuous on  and suppose that .  Then .

6. Theorem: If  and  and  and  are real numbers, then .

2.2       Solution of Initial Value Problems Using the Laplace Transform

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

2. 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:   .

  .

2.3       Convolution

1. Definition (Convolution): If f and g are defined on , then the convolution  of f with g is the function defined by  for .

2. Theorem (Convolution theorem): If  is defined, then .

3. Theorem: Let  and .  Then .

Example:   .

Determine f such that   .

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

Example: Solve   .

2.4  Unit Impulses and the Dirac’s Delta Function

1. Dirac’s delta function: where; ; .

2. Theorem (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.

Example: Solve   .

 3. Laplace Transform Solution of Systems:

      Example: Solve the system

4. Differential Equations with Polynomial Coefficients

1. Theorem: Let  for  and suppose that F is differentiable.  Then  for .

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

Example:   .

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

Example:   .

   - 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

                                                                                                              (1)

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):

.

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 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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