Abstract
In this paper, a new method based on combination of the natural transform method (NTM), Adomian decomposition method (ADM), and coefficient perturbation method (CPM) which is called “perturbed decomposition natural transform method” (PDNTM) is implemented for solving fractional pantograph delay differential equations with nonconstant coefficients. The fractional derivative is regarded in Caputo sense. Numerical evaluations are included to demonstrate the validity and applicability of this technique.
1. Introduction
Delay differential equations (DDEs) occur when the rate of change of process is specified by its state at a certain past state known as its history. These types of differential equations occur in traffic models, control systems, population dynamics, and many natural phenomena [1–3]. The expression of delay in mathematical modeling of real-life phenomena leads to a precise and acceptable description of the dynamics of the model. DDEs are complex in nature because the history of the system over the delayed interval is given as an initial condition. Due to this reason, these equations are difficult to solve analytically; hence, a numerical method is required. Pantograph delay equations are one of the well-known functional differential equations with proportional delay and often appear in many scientific models such as mechanics and electrodynamics [4, 5]. In 1851, it was the first time that a device named “pantograph” was used in construction of the electric locomotive after which this name was originated from that time. Pantograph was modeled mathematically in 1971 (see [6] for more details). Thereafter, pantograph delay equation was studied by many authors and solved by several numerical methods. The most important of them are collocation method [7], spline method [8], Runge–Kutta method [9], homotopy perturbation method [10], and Adomian decomposition method [11]. In this paper, we intend to apply the natural transform method to solve fractional delay differential equations with variable coefficients. To solve differential and integral equations, several integral transforms such as Fourier, Laplace, and Sumudu are used [12–16]. The natural transform method is a new integral transform was introduced by Khan and Khan [17], and its properties were described. Belgacem used the natural transform method to solve Bessel’s differential equation, Maxwell’s equation, and nonlinear Klein–Gordon equations (see [18, 19] and its references). A table for some natural transformation properties on certain functions was presented by Khan and Khan in [17]. Moreover, this method is applied to find the solution of diffusion equations [20]. Finally, they proved that the natural transform method converges to the Laplace and Sumudu transforms.
In this paper, a novel technique is applied to find an approximate solution for the fractional pantograph delay differential equation of the following form:where , and are the given analytic functions, and denotes the Caputo fractional differential operator. The proposed method is a combination of natural transform method (NTM), Adomian decomposition method (ADM) [21], and coefficient perturbation method (CPM) [22], which is called “perturbed decomposition natural transform method.” Unlike existing analytical and numerical methods, this method gives an approximate and/or exact solution for problem (1) by a simple calculation and in an elegant way.
The structure of this paper is as follows: In Section 2, a brief description of the fractional calculus, natural transform method, and Adomian decomposition method are provided. The existence of unique solution for problem (1) is investigated in Section 3. In Section 4, we introduce our new method (called briefly, PDNTM) for obtaining the approximate solution for problem (1). Numerical examples illustrate efficiency and applicability of the proposed method in Section 5, and conclusions are drawn in Section 6.
2. Preliminaries
In this section, we recall some preliminary results which will be needed throughout the paper.
2.1. Fractional Calculus
There are some kinds of definitions for fractional derivatives and integrals, such as Riemann–Liouville integral [23], which are described as follows.
For , the Riemann–Liouville fractional integral of order α is defined as
For , set , the identity operator. Let for some and . Then,
Let . The operator defined byis called the Riemann–Liouville fractional differential operator of order α. For , set , the identity operator. The Caputo differential operator of order n is defined bywhenever exists, where denotes the Taylor polynomial of degree for the function f around the point In the case , define . Under the above conditions, it is easy to show that
More details can be known by referring to [23, 24].
2.2. Natural Transform Method (NTM)
In this section, we review some properties of the atural transform method that will be used in later sections. For the function defined in , piecewise function continues in every finite interval and whereThe natural transform of is defined as follows:With the above conditions, the right-hand side integral in (39) is convergent. By taking in (39), the Laplace transform of ,and for , the Sumudu transform of ,will be obtained. The inverse natural transform of is defined in [19] by
We recall some useful properties of the natural transform method from [19] in Table 1.
2.3. Adomian Decomposition Method (ADM)
Consider the general form of a differential equation aswhere is the linear invertible part, corresponds to the nonlinear term, and is the remaining part. Solving from (12), we have
As L is invertible,
The Adomian decomposition method (ADM) [21] is a powerful tool for solving both linear and nonlinear functional equations. This method decomposes the solution into a rapidly convergent series of solution components and then decomposes the analytic nonlinearity into the series of the Adomian polynomials:where are the well-known Adomian polynomials as follows:
Substituting (15) in (14), we have
Consequently, we write
The convergence analysis of the Adomian decomposition method was studied by Cherruault in [25, 26].
3. Existence and Uniqueness of Solution
In this section, we prove existence and uniqueness of the solution for the fractional delay differential equation (1). This study is based on the Banach contraction principle.
Theorem 1. Assume and are the given analytic functions in (1) and . Ifthen problem (1) has a unique solution .
Proof. The equivalent integral equation representation of problem (1) is of the following form (see [24]):Based on the Banach contraction principle, it is enough to prove that the operator A defined byhas a unique fixed point. It is clear that the operator A maps into itself and thatwhich implies, under our assumption, that A is a contraction mapping. Then, the Banach contraction principle implies that A has a unique fixed point .
4. Main Results
Let us approximate the coefficients and , in equation (1) by polynomials of fractional order α as follows:
Then, problem (1) can be written in perturbed form aswhich is called “perturbed problem” with the exact solution that is an approximation for the exact solution of the main problem (1). This is a first step of our new method that converts the main problem (1) to new one with polynomial coefficients. This strategy will simplify the calculation in the implementation of the proposed method. Similar ideas have been used in [27, 28] for solving boundary value problem and eigenvalue problems.
Theorem 2. Let andfor some constant in (23). Then, we have , i.e.,
Proof. It can be easily proved that is a solution of equation (1) if and only if is a solution for the integral equationwhere is the Riemann–Liouville fractional differential operator of order α (see [24]). Consequently, for perturbed problem (24), we haveSubtracting equation (28) from equation (27) and some manipulation yieldBy taking from both sides of equation (29), we obtainFrom (25), we haveThen, from (30) and (31), we getConsequently,After some manipulation on (33), we obtainFrom the upper bound in (34), it is seen that if , then .
4.1. Method of Solution
Taking the natural transform of both sides of equation (24) yields
Using the properties of the natural transform method in Table 1, we get
Consequently,
By taking the inverse natural transform of (37), we obtain
Now, based on the Adomian decomposition method, we construct the solution in (38) by an infinite convergent series of functions as follows:
Substituting (39) in (38) yieldsin which the components will be obtained by the following recurrence relation:such that
The -term approximation of the solution isso that
Therefore, the solution comes in the form of a convergent series with easily computed components. From Theorem 2 and (23), we know that as .
Remark 1. As is clear, in recurrence relation (41), the terms and can be calculated easily using properties (3) in Table 1, since the term is a fractional-order polynomial for and . This is a main advantage in our proposed method for calculating the components in (43) and making our new method easy to be used and writing a computer program for it.
We summarize the steps of our proposed method (called PDNTM method) as an implementation algorithm as follows (Algorithm 1).
5. Numerical Results
In this section, we present some examples to support the accuracy and simplicity of the proposed method. These results in this section indicate the applicability of the method and verification of the theoretical results in the previous section.
Example 1. We consider the following initial value problem:The exact solution for is . According to the proposed method, by taking natural transform to both sides of equation (45), we obtainFrom initial condition , we obtainApplying the inverse of natural transform in (47) yieldsNow, based on (41), we obtainwhich givesThen, all of the components can be obtained as follows:Continuing the above recurrence relation yieldsFor the case , the second term in (52) is equal to zero; therefore, , which is the exact solution of problem (45). For , the approximate solution isIn Figure 1, the convergence of approximate solutions to exact solution as is shown.
Example 2. Consider the following fractional pantograph problem (see [29]):In the case , the exact solution is . According to the proposed method for , we obtainThen, from (43), we obtainwhich is the exact solution. The approximate solutions for different values of α and exact solution are shown in Figure 2 for . Note that as , the numerical solutions tends to the exact solution .
Example 3. We consider the following nonpolynomial coefficients problem (see [29]):The exact solution for is . LetThen, the perturbed problem is as follows:The approximate solutions for and exact solution for , , and are shown in Figure 3. Table 2 denotes the behavior of approximate solution for different values of with . These results are consistent with Theorem 2 which shows convergence of the solution of perturbed problem (59) to (57).
6. Future Work
In our future work, the PDNTM method will be studied for the general class of problems such as the system of fractional pantograph delay differential equations of the formsubject to the initial conditionswhere and . To this end, we can approximate the coefficients and by polynomials of fractional order as follows:and extend all processes of Section 4.1.
7. Conclusion
In this paper, a new method based on the combination of the natural transform method, Adomian decomposition method, and coefficient perturbation method is introduced to solve the fractional pantograph delay differential equations with nonconstant coefficients. From this method, we obtain an approximate or exact solution for the problem with some simple iterative calculation that is easy in use and computer programming. The numerical results confirm the accuracy of the method.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors appreciate the reviewer’s invaluable advice on improving the paper.