ANALYTICAL SOLUTIONS OF VOLTERRA AND FREDHOLM INTEGRAL EQUATIONS

CHAPTER ONE

                         BACKGROUND OF THE STUDY

1.1 INTRODUCTION.       

               Integral equations play an effective role in the study of boundary value problem. Such equations also occur in many fields of mechanics, chemistry, biology and mathematical physics. Integral equations may be obtained directly from physical problems for example radiation transfer problem, neutron diffusion problem e.t.c. They also arise as representation formulae for the solutions of differential equation. A differential equation can be replace by an integral equation with the help of initials and boundary condition.

              The use of differential equations in obtaining solutions to many natural and man-made phenomena has common practice for many years. However, there are situation where a differential equation cannot accurately represent the physical nature of the problem, and as a result the introduction of integral equations was realized. 

              Although work on integral equations began in the 1820’s with Abel, it is believed that Vito Volterra was first who conceived the idea of a function whose behavior depends on a continuous set of values obtained from another function, and his first paper on this subject was published in 1896, and the Swedish mathematician E.I Fredholm in 1900, which are our main consideration.

1.2 AIM AND OBJECTIVES

         The main aim of this work is to have solution of Volterra integral equations by conversion to ordinary differential equation and Fredholm integral equations with separable kernel, with the following objectives

To obtain the analytical solution of integral equations.

To solve some problems with distinct kernel.

To study integral equations in solving problems of Fredholm and Volterra equations.

1.3 SCOPE AND LIMITATION

Most of the integral equations met in practice are solved numerically with a computer, such as trapezium rule method, Simpson’s  method and the Gregory quadrature method e.t.c. But this work is restricted to solution of Fredholm integral equations with separable kernel and Volterra integral equations by conversion to ordinary differential equations.

1.4 SOME OF BASIC CONCEPT DEFITIONS.

1.4.1 INTEGRAL EQUATION        

         An integral equations is an equations in which the unknown function to be determined appear under an integral sign.                

                      U

Where k(x, y) is called the kernel

1.4.2 VOLTERRA INTEGRAL EQUATION 

Integral equation whose upper limit is x is known as Volterra integral equation. 

                               U

Where x is a variable or infinite

1.4.3 FREDHOLM INTEGRAL EQUATION

                    Integral equations whose upper and lower limits are constants or finite are known as Fredholm integral equations. 

                           U

1.4.4 LAPLACE TRANSFORM

Ifrepresent some expression in t defined for t>0, the Laplace transform, donated by, is defined to be

Where s is a complex variable and  is called the kernel of the transformation.

1.4.5 DIFFERENTIAL EQUATION

                 A differential equation is a relationship between an independent variable x, a dependent variable y, and one or more derivatives of y with respect to x.  Or 

Differential equation is an equation having an unknown function and it derivatives. For example

1.4.6 INTEGRO-DIFFERENTIAL EQUATION

                           Equation involving both derivatives and integrals of unknown variable is known as integro-differential equations. For example

                     L

1.4.7 LEIBNIZ INTEGRAL RULE

                 The Leibniz integral rule gives a formula for differentiation of a definite integral whose limits are functions of the differential variable,

It is sometimes known as differentiation under the integral sign.

A typical form of an integral equation is,                                                                                                             

Where k(x, t) is called the kernel of the integral equation and a, b are the limit of integration. It is easily observed that the unknown function u(x) appears under the integral sign in most cases.

1.5 CLASSIFICATION OF LINEAR INTEGRAL EQUATIONS

          Integral equations are classified into three categories, they are as follows:-

Limit of integration

Placement of the unknown function

Nature of the known function

1 LIMIT OF INTEGRATION

When the limits are either fixed or constant, the integral equation is known as Fredholm equation. For example

When one of the limits is a variable or infinite, the equation is known as Volterra equation.  For example

2 PLACEMENT OF THE UNKNOWN FUNCTION

When the unknown function appears only under the integral sign, then it is known as integral equation of the first kind. For example

When the unknown function appears both inside and outside of the integral sign, then it is known as equation of second kind. For example

3 NATURE OF THE KNOWN FUNCTION

When the known function  is equal to zero. Then we say that the equation is a homogeneous integral equation. For  example

When the known function  is not identically equal to zero, then it is known as inhomogeneous integral equation. For example   

CHAPTER   TWO

2.1 HISTORICAL DEVELOPMENT OF INTEGRAL EQUATIONS                                               

The integral equation was first introduced by Paul du-Bois-Rey mound in 1782 when he the integral transforms.

to solve linear difference and differential equations.  It was later worked on by poison in 1826.

       As an appetizer we consider Abel’s integral equation that occurred as one of first integral equations in mathematical history. A tautochrone is a planar curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point. The problem to identify this curve was solved by Christiaan Hygens in 1659 who, using geometrical tools, established that the tautochrone is a cycloid.

            In 1823 Niels Henrik Abel attached the more general problem determining a planar curve such that the time of descent for a given starting height y coincides with the value f(X) of a given function f. The tautochrone then reduces to the special case when f is a constant.

The above is known as Abel’s integral equation.

Early year

        We proceed by giving a brief account on the close connection of the early development of integral equations with potential theory. In the sequel x = (x1, x2) and y = (y1, y2) stand for points or vectors in the Euclidean space.  Twice continuously differentiable solution u to the Laplace equation

Are called harmonic function. They model time –independent temperature distributions, potentials, of electrostatic and magnetostatic fields, and velocity potentials of incompressible irrigational fluid flows.

        A first approached to Dinichlet problem of potential theory, developed in the early 19th century was to create a so-called single-layer potential by distributing point sources with a density on the boundary curve i.e. by looking for a solution in form

Since  satisfies Laplace’s equation if x the function u is harmonic and in order to satisfy the boundary condition it suffices to choose the unknown function  as a solution of the integral equation

This is known as Symme’s integral equation.  However, the available atthat time did not allow a successful treatment of this integral equation of first kind, only in the second half of the 20th century was a satisfying analysis of the equation achieved. Therefore, it represented a major breakthrough when August Beer in 1856 proposed to place dipoles on the boundary curve. I.e. to look for a solution in the form of a double lawyer potential.

      Now the so called jump relations from potential theory required that

However, in order to achieve convergence for the case of convex domains in 1877 Carl Neumann had to modify the successive approximation into what he called method of arithmetic means. I.e., a relaxation method in modern terms. 

   For the general case, establishing the existence of a solution to integral equation had wait until the pioneering results of Ivar Fredholm that were published in final form in 1903 in Acta mahtematica with the title Surune classe d’equations fonctionelles.  Fredholm considered equations of the form

With a general kernel k and assumed all the functions involved to be continuous and real valued.  His approach was to consider the integral equation as the limiting case of a system of linear equations by approximating the integral by Riemannian sum. In crammer’s rule for this linear system Fredholm passes to the limit by using Koch’s theory of infinite determinants from 1896 and Hadamard inequality for determinant from 1893. The idea to view integral equations as the limiting case of linear system had already been used by Volterra in 1896, but it was Fredholm who completed it successfully.

         In addition, Freehold’s results also contain the ad joint integral equation that is obtained by interchanging the variables in the kernel function. Freehold’s results on the integral equation initiated the development of modern functional analysis in the 1920s. The almost literal agreement of the Fredholm alternative for linear integral equations with the corresponding alternative for linear system soon gave rise to research for a broader and more abstract form of the Fredholm alternative. This in turn allowed extensions of the integral equation theory under weaker regularity requirements on the kernel and solution function. In addition, many years later it was found that more insight was achieved in the structure of Fredholm integral equations, by altogether abandoning the at first very fruitful analogy between integral equations and linear system.  

       A first answer to the search for a general formulation of the Fredholm alternative was given through Riesz. In his was from 1916 he interpreted the integral equation as a special case of an equation of second kind 

With a compact linear operator Amapping   a normed space X into self. The notion of a normed space that is common in today’s mathematics was not yet available in 1916.

    The concept of a compact operator also was not yet available in 1916. However, using the notion of compactness as introduced by Fredholm in 1906, Riesz formulated that the integral operator A defined by

                            (A)

On the space of continues functions maps bounded sets into relatively compact sets, I.e., in today’s terminogy A is a compact operator.   The conceptually most straight forward idea for the solution of integral equations of second kind goes back to Nystron in 1930 and consists of replacing the integral in integral equation by numerical integration. For problems in mathematics physics in 1923 Hadamard postulated three requirements. A solution should exist, the solution should be unique, and the solution should depend continuously on the data.

         Kress in 1990 introduced an algebraic transformation for smoothing the solution of boundary Fredholm integral equation in domain with corner. The solution of this integral equation has a singularity on the corner point. He considered integral equation of the second kind in the slightly unconventional form, and supposed that the input function is continuous.

2.2 A BRIEF REWIEW OF EARLIER WORKS

            A lot has been written on the application of integral equations but in this work, deliberate and conscious effort have been made to solve integral equation that occurred in science and technology produce.

           It is good to review briefly some earlier works written by some other researchers or writers on the topic.

         In the book, a first course in integral equation Abdul-majid Wazwaz, explain that every physical problem that has a differential equation, there is an equivalent integral equation. He goes further to explain that a linear differentia equation can be transformed into an equivalent integral equation and vice versa.

          To make this transformation he makes use of these formulae. 

In general

       In the book, Laplace transform and application, E.J Watsonof the Department of mathematics. University of Manchester, had shown how Laplace transformation could be applied to solve a variety of problems, especially integral equation. He goes further to say that considering: 

                               (1

                                                                                                                                       (2)

The two integral equations above are integral equations of convolution type for the unknown function U(x). They are example of Volterra equations in which the kernel function k(x, t) = k(xwhere the function k(x) and g(x) are assumed to be defined for all t>0.

          He explain further that if k(X) and g(X) have Laplace transforms, the equations above of first and second kind can be solved explicitly.

CHAPTER THREE

3.1 METHEDOLOGY

In this chapter we are going to outline some of the integral equations that occurred in mechanics and mathematical physics. We are also going to solve Volterra integral equations by conversion to ordinary differential equations, and Fredholm integral equations by separable kernel.

3.2 VOLTERRA INTEGRAL EQUATIONS

          In special cases a Volterra integral equation can be solved by conversion to an ordinary differential equation. Taking our standard linear Volterra integral equation of second kind, thus

1

And differentiating both side with respect to t (assuming that  and k(t, s) are differentiable),

)

We apply Leibniz rule of differentiation upon an integral to form.

                     2

Rearranging equation (1), to be,

And

Then we the substitute (3) in (2) to get

This ordinary differential equation can then be solved to find a solution for f(t), where f(0)}. If, however (4) does not hold, the equation (3) can be differentiated again and substitution made if, (assuming second derivative of k(t, s) exist)

With initial value of  and

In general a linear Volterra equation of the second kind may be converted to an ordinary differential equation and then solved if 

For some n

   Examples

3.2.1 Solve the Volterra integral equation of

Solution

Substitution expression is                                                                                                

Differentiating both side with respect to x

We apply leibniz rule of differentiation on integral

 We differentiate again

By apply Leibniz rule of differentiation upon an integral we have,

               But   

We now solve the differential equation with initial value y(0) = 0  and y’(0) = 0. Taking the Laplace of both sides we have,

Putting our initial value we have,

Taking the inverse of Laplace transform we have,

  +

3.2.2 Consider the equation

Solution

Our substitution expression is

Differentiation of the equation yield, 

Applying Leibniz rule differentiation upon integral we get, 

Differentiate second time again gives,

The equation can be converted to ordinary differential equation, which has the solution (when 

Taking the Laplace transform of both sides of the equation we have,

()

Now taking the inverse of the Laplace transform gives,

Splitting into partial fraction we get,

                                                                               1

                            2   

.                                            Constant:

From (3) 

Putting the value of B into (2) we have,

Putting the value A into (1) yield,

3.2.3 Solve the integral equation

Solution

  Our substitution expression is,

Differentiation of the equation above we have,

Differentiating second time gives,

The equation above can be converted to ordinary differential equation, which has solution when 

   Putting the initial values we have,

 ,     

Taking the inverse of Laplace transform we get,

3.3 FREDHOLM INTEGRAL EQUATIONS 

      A kernel k(x, t) of an integral equation is said to be seperable if it can be expressed as the sum of finite number of terms each of which is the product of a function of x alone and function of t alone if it is of form.

When the kernel k(x ,t) is written as

       K(x ,t) = g(x)h(t) 

Substituting  into Fredholm integral equation we have,

         Let                   C

Examples

3.3.1 Consider the equation

Solution

Let 

Replace x by y we have,                                                                                  

.

3.3.2 Solve the equation by separating kernel.

Solution

The kernel is 

We have, by separating the kernel.

Let 

Thus 

So,                                                    1

By putting (1) into c1 and c2 we have,

2

                                                     3

Now, solving (2) and (3) simultaneously gives,

From (3)   

Putting (4) into (3) we have,

By substituting c2 into equation (4) we have,

By putting the values of c1 and c2 into equation 1 we have,

3.3.3 Find the eigen value and eigen function of Fredholm integral equation

                                                                       1

Solution

The given integral equation can be rewritten as,

Substituting y(t) into c1 and c2 we have,

Then, the determinant of eigan value will be,

Putting  in equation (2) yield,

Hence 

From equation (1)

Therefore, the eigen functions corresponding to the eigen value is

CHAPTER FOUR

SUMMARY AND CONCLUSION

4.1 SUMMARY

In summary, this project work analyzes the two types of integral equations called Volterra and Fredholm integral equations. The following are discussed, back ground of the study where we establish an introduction, some basic concept definitions and classification integral equation into three categories.

Limit of the integration

Nature of the unknown function

Placement of the known function

      We discussed the historical development of the Fredholm integral equation and Volterra integral equation. In which the Volterra integral equation is given by

Where  is called kernel, which by conversion to ordinary differential equation solved some problems and obtained solutions of Volterra integral equations. In other way the Fredholm integral equation is given by,

This shows that Volterra and Fredholm integral equations can only be differentiated by their upper limit i.e. Volterra its upper limit is a variable while Fredholm has constant.    

    4.2 CONCLUSION

In conclusion, we should be able to differentiate between Volterra integral equations and Fredholm integral equations, also to solve Volterra integral equation by conversion to ordinary differential equation and Fredholm integral equation by seperable kernel.

      From the problems we have solved, it can be understand that most of the solutions are in form of polynomial equations and also, application of mathematics in physics enables us to comprehend concept that are otherwise not clearly understandable for example Fredholm equations, Volterra equations, Abel´s equations e.t.c that are always associated with problems in physics. 

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