11.3 Nonhomogeneous Boundary Value Problems

We discuss how to solve nonhomogeneous boundary value problems in this section.

Problems in which the differential equation alone is nonhomogeneous are the ones that get the most attention.
We assume that the solution can be expanded in a series of eigenfunctions and that the coefficients can be determined so that the nonhomogeneous problem is satisfied.
This method is used for boundary value problems for second order linear ordinary differential equations.
We show its use for partial differential equations when we solve a heat conduction problem in a bar with variable material properties and in the presence of source terms.

This is just an example.

The nonhomogeneous term is suggested to be changed in Eq.

One of the main cases has two sub cases.

Our argument doesn't show that the series converges.

The solution of the boundary value problem clearly complies with the more stringent conditions given in the paragraph following that theorem.

We have to consider two cases again.

It is not possible to solve Eq.

The results we have formally obtained are summarized.

The Fredholm4 alternative theorem is the last form of the theorem.

In many different contexts, this is one of the basic theorems of mathematical analysis.
There is a discussion in Section 7.3.

eigenfunction expansions can be used in many problems that are not accessible by elementary procedures.

To identify it.

The theory of integral equations was established in 1903 by the professor at the University of Stockholm.

The similarities between integral equations and systems of linear equations were emphasized by Fredholm's work.
There are many interrelations between differential and integral equations.

The homogeneous problem has an eigenvalue of 2.

One can show the equivalency of Eqs.

We emphasize again that we may not be able to get the solution by series or numerical methods.

The problem was discussed in Appendix A of Chapter 10.

We substitute from the beginning.

Consider the term on the left side.

The nonhomogeneous term must also be expressed.

We substitute from the results.

To make it simpler.

This is obtained from the initial condition.

The methods of Section 2.1 were used to solve the initial value problem.

You can leave the details of this calculation to us.

An explicit solution of the boundary value problem is given.

To use this method to solve a boundary value problem.

The integral is evaluated.

The process can be difficult since any of the steps may be difficult.

A very few terms may be needed to get an adequate approximation to the solution if the series converges rapidly.

The solution was given by Eq.

We must estimate the speed of convergence to judge whether a sat isfactory approximation to the solution can be obtained.
We split the right side of the book.

All terms after the first are insignificant.

The preceding discussion and examples may suggest that igenfunction expansions can be used to solve a lot of problems.

The procedure described in this section can be used to solve this problem.

The procedure for dealing with this type of problem is more complicated and we will not discuss it.

The normalized eigenfunctions of the corresponding problem must be found in order to use eigenfunction expansions.

This may be difficult for a differential equation with variable coefficients.
Sometimes it is possible to use other functions that satisfy the same boundary conditions.

Although they are not solutions of the corresponding differential equation, these functions at least satisfy the correct boundary conditions.

The infinite system is replaced by an approximating finite system, from which approximations to a finite number of coefficients are calculated.

This procedure has proved to be very useful in resolving difficult problems.

Boundary value problems can be solved by eigen function expansions.

The procedure is almost exactly the same for second order problems.
There are a variety of problems that can arise, and we will not discuss them in this book.

The discussion in this section has been formal.

To justify some of the steps used, such as term-by-term differentiation of eigenfunction series, separate and sometimes elaborate arguments must be used.

Different methods are used to solve nonhomogeneous boundary value problems.

One leads to a solution that is a definite integral rather than an infinite series.

Nonhomogeneous problems related to the wave equation or its generalizations can be solved using the method of eigenfunction expansions.

The problem can arise in connection with generalizations of the telegraph equation or the longitudinal vibrations of an elastic bar.

There is an analogy between the problem of ary value problems and the problem of Hermitian matrices.

We will point out a way to solve it.

He made significant contributions to electricity and magnetism, fluid mechanics, and partial differential equations, and was almost entirely self-taught in mathematics.

An essay on electricity and magnetism was his most important work.
Green was the first to recognize the importance of potential functions.
He introduced the functions now known as Green's functions as a means of solving boundary value problems, and developed the integral transformation theorems of which Green's theorem in the plane is a particular case.
Green's essay was published in the 1850s through the efforts of William Thomson.

The inverse of the matrix of coefficients is played by the Green's function.

The method leads to solutions that are definite integrals.

The usefulness of a Green's function solution is dependent on the fact that the Green's function is independent of the nonhomogeneous term in the differential equation.

Again, we see from Eq.

Some of the equations of physical interest are not satisfied.

The conditions imposed on regular problems are met elsewhere in the interval.

One or both of the functions fails to satisfy them at one or both of the boundary points.

We prescribe separated boundary conditions of a kind that will be described later in this section.
This is not a singular problem in the book.

The equation is related to the study of free vibrations of a circular elastic membrane.

Bessel's equation of order zero is Equation 8.

We might try to find a solution of Eq.

It's too restrictive for the differential equation.

It is necessary to consider a modified type of boundary condition at a singular boundary point.
In the present problem, we only need that the solution (9) and its derivative remain bound.

It is possible to show seven.

The problem is given.

The boundary value is an example of a problem.

This example shows that if the boundary conditions are relaxed in an appropriate way, a single Sturm-Liouville problem may have an infinite sequence of eigenvalues and eigenfunctions.

It is worthwhile to investigate singular boundary value problems a little further because of their importance.

Two main questions are of concern.

We look at the conditions under which this relation holds for singular problems.

What kinds of boundary conditions are allowed at a single boundary point is our main object.

If that point is a singular boundary point, that's zero.

They satisfy a boundary condition of the form at each regular boundary point.

There is a singular boundary point.
There is a singular boundary point.
If at least one boundary point is singular, the differential equation (1), together with two boundary conditions of the type just described, are said to form a For example.
It is clear that Eq.

In a singular problem the eigenvalues may not be discrete, which is the most striking difference between regular and singular.

It is1-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-6556 It is possible that there is only a single set of eigenvalues.

This is true of the problem with Eqs.
It is difficult to determine which case occurs in a given problem.

The methods presented in the book need to be extended in order for a systematic discussion of the problems to be understood.

There is an infinite set of eigenvalues in each of the examples that we restrict ourselves to.

There is only a single set of eigenvalues and eigenfunctions in the problem.

Section 11.2 states that the expansion of a given function in terms of a series of eigenfunctions follows.

In the regular case, such expansions are useful for solving nonhomogeneous boundary value problems.

The procedure is the same as described in Section 11.3.
Some problems for partial differential equations can be found in Section 11.5.

The case is covered by the convergence of the series.

It can be shown to hold for other sets of Bessel functions, which are solutions of appropriate boundary value problems, and for solutions of a number of other Sturm-Liouville problems of interest.

The problems mentioned here are not typical.

In general, singular boundary value problems are characterized by continuous spectrums.
The series expansions of the type described in Theorem 11.2.4 do not exist, because the corresponding sets of eigenfunctions are not denumerable.
Appropriate integral representations are used to replace them.

There is an infinite sequence of these roots.

There is an infinite sequence of zeros.