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An analytical-approximate method is proposed for a type of nonlinear Volterra partial integro-differential equations with a weakly singular kernel. This method is based on the fractional differential transform method (FDTM). The approximate solutions of these equations are calculated in the form of a finite series with easily computable terms. The analytic solution is represented by an infinite series. We state and prove a theorem regarding an integral equation with a weak kernel by using the fractional differential transform method. The result of the theorem will be used to solve a weakly singular Volterra integral equation later on.

Many engineering and physical problems result in the analysis of the nonlinear weakly singular Volterra integral equations (WSVIEs). These equations are applied in many areas [

The aim of this paper is applying the fractional differential transform method (FDTM) for solving WSVIE. The fractional differential transform method has recently been developed for solving the differential and integral equations. For example, in [

The main challenge of partial integro-differential equations (PIDEs) with a weakly singular kernel is faced when we are looking for an analytical solution. By applying the differential transform method, the result mostly obtained is an analytical solution in the form of a polynomial. The differential transform method is different from the traditional high order Taylor series method, which requires symbolic competition of the necessary derivatives of the data functions. Making use of this method enables us to obtain highly accurate results or exact solutions for a partial integro-differential equation. The use of application of DTM and FDTM does not require linearization, discretization, or perturbation in contrast to the methods discussed in the literature [

The form of WSVIE that we will consider in this paper with FDTM is

The numerical treatment of (

The paper is organized as follows: In Section

There are different kinds of definitions for the fractional derivative of order

There are some approaches to the generalization of the notion of differentiation to fractional orders. According to the Riemann-Liouville formula, the fractional differentiation is defined by (

Consider a function

Suppose that

if

if

if

if

if

if

if

where

If

The Beta function

The Kronecker delta function is given by

Now, we represent the main theorem of this study, through which a weakly singular Volterra integral equation can be expressed as a series of fractional differential transform for

Suppose that

By putting

In this section, we try to describe the FDTM for (

In this section, we take some examples to clarify the advantages and the accuracy of the fractional differential transform method (FDTM) for solving a kind of nonlinear partial integro-differential equation with a weakly singular kernel. For each of these examples, we obtain a recurrence relation. In all of the examples, we choose

Consider the following nonlinear partial integro-differential equation with a weakly singular kernel with

For solving (

By using the recurrence relation (

and by applying the same calculations, the following can be concluded:

Also we put

By Definition

And by applying the same calculations, we can conclude that

By continuing this process, we can also conclude the following:

Therefore, by substituting the above values into (

which is the particular solution obtained in [

Consider the following nonlinear partial integro-differential equation with a weakly singular kernel:

Taking into consideration the two-dimensional transform for (

Consider the following nonlinear partial integro-differential equation with a weakly singular kernel:

To solve (

Of course this solution is an analytical solution.

In this paper, we have described the definition and operation of two-dimensional fractional differential transform; fractional derivatives have been considered in the Caputo and Riemann-Liouville sense and the main theorem on fractional differential transform method. Using the fractional differential transform method, a kind of nonlinear partial integro-differential equation with a singular kernel was solved approximately and analytically. We have used FDTM in this paper to solve (

The authors declare that there are no conflicts of interest regarding the publication of this paper.