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
(1)
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
(2)
and
(3)
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
(4)
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
(5)
called -regressive. In addition, if is ld-continuous, then the first order inhomogenous linear dynamic equation
(6)
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
(7)
for
Theorem 3. Suppose (5) is -regressive and fix Then is the unique solution of the IVP
(8)
Next theorem gives some properties of the nabla exponential function, can be found in [3,4].
Theorem 4. Let , and Then
(i) and
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
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
(9)
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
(10)
where and are given functions that at least ld-continuous on Rewriting the integral part of Equ. (3.1) as
(11)
Thus Equ. (10) can be written as
(12)
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
(13)
Therefore, Equ. (10) can be approximated by the following first-order linear dynamic equation
(14)
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
and
which is coincide with the section 2.1 of [5].
For the points we can calculate and as
and
By the help of Theorem 5 we can write the solution of the Equ. (14) of the form
(15)
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
(16)
where is analytic in the solution
We have by setting
(17)
as before, we write this equation in the following form
or
.
An approximate solution can be found by considering
(18)
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
(19)
where and are given functions as in previous section. If we use the same procedure as in previous section we can write
or
where as in section 3.
Thus we can rewrite Equ.(3.10) as
(20)
By setting in Equ. (20) we get first-order linear dynamic equation as
(21)
where
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
(22)
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].
Acknowledgements
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.
References