International Journal of Discrete Mathematics
Volume 1, Issue 1, December 2016, Pages: 20-29

An Analytic Approach to Weakly-Singular Integro-Dynamic Equation on Time Scales

Adil Mısır

Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, Turkey

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To cite this article:

Adil Mısır. An Analytic Approach to Weakly-Singular Integro-Dynamic Equation on Time Scales. International Journal of Discrete Mathematics. Vol.1, No. 1, 2016, pp. 20-29. doi: 10.11648/j.dmath.20160101.14

Received: December 12, 2016; Accepted: December 22, 2016; Published: January 16, 2017

Abstract: In this paper, we present a new and simple approach to resolve linear and nonlinear weakly-singular Volterra integro-dynamic equations of first and second order on any time scales. In order to eliminate the singularity of the equation, nabla derivative is used and then transforming the given first-order integro-dynamic equations onto an first-order dynamic equations on time scales. The validity of the method is illustrated with some examples.

Keywords: Time Scales, Integro-Dynamic Equations, Volterra Integro-Differential Equation

1. Introduction

Linear and nonlinear Volterra integro-differential equations play an important role in mathematical modeling of many physical, chemical and biological phenomena in which it is necessary to take into account the effect of past history. Particularly in such field as heat transfer, nuclear reactor dynamics, dynamics of linear viscoelastic materials with long memory and thermoelectricity, optics, electromagnetics, electrodynamics, chemistry, electrochemistry, fluid flow, chemical reaction, population dynamics, statical physics, inverse scattering problems and many other practical applications.

During the last decades the researchers are considered the two of the most important types of mathematical equations that have been used to mathematically describe various dynamic procedure. One of them is differential and integral equations and the other is difference and summation equations, which model phenomena respectively: in continuous time; or discrete time. The researchers have used either differential and integral equations or difference and summation equations- but not a combination equations of the two areas to describe dynamic models.

Recently, it is now becaming apperent that certain phenomena do not involve only continuous aspect or only discrete aspects. Rather, they feature elements of both the continuous and discrete . These type of mixed processes can be seen, for example, in population dynamics where non-coincident generations [14] occurs. Additionaly, neither difference nor differential equations give a appropriate description of most population growth [9].

Some problems of mathematical physics are described in terms of nth-order linear and nonlinear Volterra integro-differential equation of the form


whereis the unknown function and  is the kernel of integral equations in [1, 17].

In continuous case equations of this form with degenerate, difference and symmetric kernels have been approached by different methods including piecewise polynomials [6], the spline collocations method [7], the homotopy perturbation method [16], Hear wavelets [10], the wavelet-Galerkin method [12], the Tau method [8], Taylor polynomials [11] , the sine-collocations method [19], and the combined Laplace transforms-adomain decomposition method [18] to determine exact and approximate solutions. But if Equ. (1) is weakly-singular Volterra integro-differential equations there is still no viable analytic approach for solving Equ. (1). Recently in [5] the authors are considered the approximate solutions of a class of first and second order weakly-singular form of Equ. (1) with kernel  is singular as  where  and in [15] D. B. Pachpatte gives an approximate procedure for first order dynamic integro-differential initial value problem.

In discrete case to our knowledge there isn’t any analytic approaching method to the corresponding form of Equ. (1) with weakly singular kernel to discrete form and the time scale calculus is developed mainly to unify differential, difference and  calculus. Thus in this paper we are considered the first-order linear Volterra integro-dynamic equations in any time scales and we give an approaching method to the solution of the considered integro-dynamic equations with weakly singular kernel.

2. Some Preliminaries

The calculus of time scales was introduced by Aulbach and Hilger [2] in order to create a theory that can unify and extend discrete and continuos analysis.

Definition 1. A time scale , which inherits the standard topology on  is an arbitrary nonempty closed subset of the real numbers.

Example 1. The real numbers  the integers  the natural numbers  the non-negative integers  the numbers  where  is a fixed real number, the  numbers  where  is a fixed real number, and  are examples of time scales.

In [2] Aulbach and Hilger introduced also dynamic equations on time scales in order to unify and extend the theory of ordinary differential equations, difference equations and quantum equations ( difference and  difference equations based on  calculus and  calculus). For a general introduction to the calculus on time scales we refer the reader to the textbooks by Bohner and Peterson [3, 4]. Here we give only those notions and facts concerned to time scales which we need for our purpose in this paper.

Any time scale  is a complete metric spaces with the metric (distance)  for  Consequently, according to the well-known theory of general metric spaces, we have for  the fundamental concepts such as open balls (intervals), neighborhood of points, open set, closed sets, and so on. Also we have for function  the concept of the limit, continuity and properties of continuous functions on general complete metric spaces (note that, in particular, any function  is continuous at each point of ). In order to introduce and investigate the derivative for a function  forward and backward operators play important roles.

Definition 2. For  the forward jump operator  and backward operator  is defined by respectively as follows




These jump operators enable us to classify the points  of a time scale as right-dense, right-scattered, left-dense, and left-scattered depending on whether  ,   respectively, for any . If sup< and sup is left-scattered we let  Otherwise, we let  Similarly if  has a right-scattered minimum, we let  otherwise, we let  Finally, the graininess functions   are defined by


Example 2. If  then  and  If  then   and  If  then , ,  and

Definition 3. For  and , we define the nabla derivative of  at , denoted , to be number (provided it exists) with the property that given any , there is a neighborhood  of  such that

for all .

The following theorems delineate several properties of the nabla derivative; they are found in [3, 4].

Theorem 1. Assume  is a function and . Then:

(i) ) If  is nabla differentiable at , then  is continuous at .

(ii) If  is continuous at  and  is left-scattered, then  is nabla differentiable at  with

(iii) If  is left-dense, then  is nabla differentiable at  iff the limit

exists as a finite number. In this case

(iv) If  is nabla differential at , then

Theorem 2. Assume  are nabla differential at . Then:

(i) The sum  is nabla differentiable at  with

(ii) The product  is nabla differentiable at  with

(iii) If ,then  is nabla differentiable at  with

Example 3. If  we have , the usual derivative, and if  we have the backward difference operator,

Definition 4. A function  is left-dense continuous (or ld-continuous) provided it is continuous at left-dense points in  and its right-sided limits exists (finite) at right-dense points in

Definition 5. Assume  is a regulated function. A function is called an antiderivative of  provided  for all  In this case we define the nabla integral by

We now state some definitions and at goal we will define a function , called nabla exponential function, which solves the general first order linear nabla-dynamic IVP.

Definition 6. Let be a time scale. We say that a function is -regressive provided

Define the -regressive class of functions on  to be

If , then the first order linear dynamic equation


called -regressive. In addition, if  is ld-continuous, then the first order inhomogenous linear dynamic equation


called -regressive. If , then we define the circle plus and minus by

Definition 7. For  let  and  Define -cylinder transformation  by  where  is the principal Logarithm function. For  we define  for all . If , then we define the nabla exponential function by



Theorem 3. Suppose (5) is -regressive and fix  Then  is the unique solution of the IVP


Next theorem gives some properties of the nabla exponential function, can be found in [3,4].

Theorem 4. Let , and  Then

(i)  and








Example 4. It is clear that , where  is constant, for  Now let  for  Let  be a constant, i.e.,  Then

Theorem 5. [3] Suppose (6) is -regressive. Let  and . The unique solution of the IVP


is given by

3. Solutions by Approximation Method

We start this section with the recalling the concept of an approximate method for solving linear and nonlinear weakly-singular Volterra integro-dynamic equations as in [14]. This concept will help us to constract the approximation solution of first-order nonlinear weakly-singular Volterra integro-dynamic equations and second-order linear weakly-singular Volterra integro-dynamic equations on time scales, which will be given in subsection 3.1 and 3.2 respectively.

Consider the following first-order linear weakly-singular Volterra integro-dynamic equation


where  and  are given functions that at least ld-continuous on  Rewriting the integral part of Equ. (3.1) as


Thus Equ. (10) can be written as


If we use the fact  we can take the fraction  in the second integral of Equ. (12) as approximately  Substituting the approximate relation into the right side of Equ. (10) we can get


Therefore, Equ. (10) can be approximated by the following first-order linear dynamic equation


Note that if  than Equ. (14) becomes first-order linear differential equation  and the general solution may be readily written as . Moreover for  we can calculate  and  as


which is coincide with the section 2.1 of [5].

For the points  we can calculate  and  as


By the help of Theorem 5 we can write the solution of the Equ. (14) of the form


under the initial condition  for  [3].

Theorem 6. [14]Let  and  are given functions as in Equ. (10) and  be the solution of Equ. (14) under the condition . Then  can be taken the approximate solution of Equ. (10)with the error

Remark 1. If we take  and  then the solution of Equ. (14) under the condition  will be exact solution of Equ. (10).

Example 5. Let   and . Then for  we get

respectively. Thus from Theorem 6 we find that for

as an error. For  we find that

respectively and the error will be approximately

for . Finally if we choice  and  we get

respectively and the error will be approximately

for .

3.1. First-Order Nonlinear Weakly-Singular Volterra Integro-Dynamic Equations

In this subsection we consider the first-order nonlinear Volterra integro-dynamic equation of the form


where  is analytic in the solution

We have by setting


as before, we write this equation in the following form



An approximate solution can be found by considering


where  and  are the same as before.

Therefore, Equ. (16) can be approximated by the first-order nonlinear dynamic equation Equ. (18).

Naturally Equ. (16) and Equ. (18)coincide with Equ. (2) and Equ. (8) of [5] when

3.2. Second-Order Linear Weakly-Singular Volterra Integro-Dynamic Equations

The same procedure can be adopted to transform a second-order linear weakly-singular Volterra integro-dynamic equation into an first-order dynamic equation, which permits convenient resolution of these equations.

Consider the following second-order linear weakly-linear Volterra integro-dynamic equation


where   and  are given functions as in previous section. If we use the same procedure as in previous section we can write


where  as in section 3.

Thus we can rewrite Equ.(3.10) as


By setting  in Equ. (20) we get first-order linear dynamic equation as



Therefore, Equ.(3.10) can be approximated by the first-order nonlinear dynamic equation Equ. (21).

If we use the same procedure of the Theorem 6 we get the error function as


Example 6 Let  and  where  is an abritrary constant. Then we find that

If we use these facts in Equ. (22) we find that

It is easy to sea that  for ,  for  and ,  for  and  and finally  for  and  Naturally as  increases  increases.

Note: In order to calculate  in the above examples Maple 13 software has been used.

4. Some Remarks

We have reduced the solution of a class of linear and nonlinear weakly-singular Volterra integro-dynamic equations to the solution of ordinary dynamic equations by removing the singularity using an approximate nabla derivative. Then we have demonstrated the solution of these ordinary dynamic equations, which approximate the solution for the original weakly-singular Volterra integro-dynamic equations.

5. Conclusions

We have considered several distinct examples to illustrate our new approach and have verified our solution, beginning with first-order and second order linear weakly-singular Volterra integro-dynamic equations. Of course, it would be better to obtain a similar procedure if  is an arbitrary ld-contunious function in Equ.(3.10). It seems that ones can get over the problem by using the Taylor expansions of a function on time scales [3, 4].


The author would like to express his sincere gratitude to the referees for a number of valuable comments and suggestions which led to signi.cant improvement of the final version of the paper.


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