#### Abstract

Some researchers have combined two powerful techniques to establish a new method for solving fractional-order differential equations. In this study, we used a new combined technique, known as the Elzaki residual power series method (ERPSM), to offer approximate and exact solutions for fractional multipantograph systems (FMPS) and pantograph differential equations (PDEs). In Caputo logic, the fractional-order derivative operator is measured. The Elzaki transform method and the residual power series method (RPSM) are combined in this novel technique. The suggested technique is based on a new version of Taylor’s series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates computing the fractional derivatives each time. As ERPSM just requires the concept of a zero limit, we simply need a few computations to get the coefficients. The novel technique solves nonlinear problems without the need for He’s and Adomian polynomials, which is an advantage over the other combined methods based on homotopy perturbation and Adomian decomposition methods. The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the presented method. Graphical significances are also identified for various values of fractional-order derivatives. As a result, the procedure is quick, precise, and easy to implement, and it yields outstanding results.

#### 1. Introduction

Many differential equations (DEs) that arise in applications are sufficiently complicated that closed-form solutions are not always feasible. Numerical methods offered a powerful substitute means for solving the DEs under the given initial conditions. Numerous methods have been developed in recent years to solve fractional-order differential equations (FODEs), including the homotopy perturbation method [1], the differential transform method [2], the operational matrix method [3], the conformable Shehu transform decomposition method [4], the variational iteration method [5], the Jacobi collocation method [6], the conformable Shehu transform iterative method [7], the spectral tau method [8], the Legendre wavelet method [9], the fractional natural decomposition method [10], the power series method with the conformable operator [11], and the Chebyshev polynomial method [12].

Integral equations (IEs), DEs, and delay differential equations (DDEs) are all solved by employing integral transforms [13–17], which are among the most valuable techniques in mathematics. The conversion of DEs and IEs into terms of a simple algebraic equation is enabled by the appropriate selection of integral transform. The origins of integral transforms can be traced back to Laplace’s work in the 1780s and Fourier’s work in 1822 [18]. In the beginning, ordinary and partial DEs were solved using the Laplace transform and the Fourier transform, which are two well-known transforms. These transforms were then applied to FODEs [19–24]. In recent years, researchers have proposed lots of new different transformations to solve a variety of mathematical problems. FODEs are solved using the Aboodh transform [25], fractional complex transform [26], travelling wave transform [27], Sumudu transform [28], and ZZ transform [29]. These transformations are paired with additional analytical, numerical, or homotopy-based techniques to handle FODEs [30–35]. Numerous mathematicians have recently become interested in a transformation known as the Elzaki transform (ET) [36–41]. The ET was introduced by Elzaki to facilitate the process of solving ordinary and partial DEs in the time domain [42]. The ET is derived from the classical Fourier integral transform.

We examine the functions in set , which are described as

The formula for E-T is as follows:

The following are the key advantages of the ET [36–45]: (i)The ET can easily be applied to the initial value problems with less computational work(ii)The ET has unit-preserving properties and may be used to solve problems without resorting to the frequency domain(iii)Numerous nonlinear DEs with variable coefficients, namely, the time-fractional wavelike equations, can be solved with it(iv)It may handle a variety of difficult problems in engineering, physics, fluid mechanics, chemistry, and dynamics, such as Maxwell’s equations and fluid flow problems

The Jordanian mathematician, Arqub, created the RPSM in 2013 [46]. The RPSM is a semianalytical method; it is a combination of Taylor’s series and the residual error function. It provides series solutions of linear and nonlinear DEs in the form of convergence series. In 2013, RPSM was implemented for the first time to find solutions to fuzzy DEs. Furthermore, this method has been successfully used to solve a wide range of FODEs, including time-fractional KdV-Burgers equations [47], time-fractional Schrödinger equations [48], the SIR epidemic model of fractional order [49], conformable-type Coudrey–Dodd–Gibbon–Sawada–Kotera [50], time-fractional Swift–Hohenberg problems [51], time-fractional Phi-4 equation [52], and the Zakharov–Kuznetsov equation [53].

Researchers combined two powerful methods to develop a new method for solving FODEs. Some of these groups are described as a combination of the Adomian decomposition method and the Sumudu transform [54], as well as the homotopy analysis method and the natural transform [55] and the Laplace transformation with homotopy perturbation approach [56]. In this study, we applied the novel combined technique, known as the ERPSM, to provide approximate and exact solutions for FMPS and PDEs. To assess the efficiency and consistency of the proposed method, the relative, recurrence, and absolute errors of the problems are examined. Graphical significance is also found for various values of fractional-order derivatives. As a result, the technique is rapid, precise, and simple to use, and it produces excellent results. The set of rules for this new technique depends on transforming the given equation into the ET space, in the second step; establishing a series solution by using the new form of the Taylor series; and then acquiring the solution in the real space of the equation by applying the inverse ET.

This novel technique can be used to construct power series expansion solutions for linear and nonlinear FODEs without perturbation, linearization, or discretization. Unlike the classical power series method, this method does not need to match the coefficients of the corresponding terms, and a recursion relation is not needed. The new method handles nonlinear problems without the need for He’s and Adomian polynomials, which is an advantage over existing combination methods based on homotopy perturbation and Adomian decomposition methods. This technique finds the coefficients of the series, relying on the limit concept but not the fractional derivatives as in the RPSM. Thus, only a few calculations are required to determine the coefficients related to RPSM. The closed-form and approximate solutions can be obtained by the proposed method through a quick convergence series.

A PDE is a special kind of delay differential equation (DDE) with a proportional delay. In 1851, the first-time device named “pantograph” was used in the construction of an electric locomotive, which is where this name originated from that time. British Railways decided to make a new kind of electric locomotive in 1960. The target was to construct a new kind of electric locomotive that moves trains faster. The pantograph was a prominent part of the new fast-speed electric locomotive. Pantographs take current from an overhead wire, which is necessary for the locomotive to move. Therefore, Ockendon and Taylor observed the mechanism of the pantograph. As a result, they built a special kind of DDE form: where are real constants and , This article was first published in 1971 [57]. Then, such a type of DDE was named PDE. Various studies on PDEs have recently been published in the scientific literature [58–60]. PDEs are widely used in probability theory, nonlinear dynamical systems, astrophysics, quantum mechanics, electrodynamics, and cell growth [61–64]. In this study, we consider the following FMPS: subject to the initial conditions where are finite constants; are analytical functions; and is the Caputo fractional derivative (CFD). The FMPS is a type of DDE that arises in a number of physics and engineering applications, including quantum mechanics, dynamical systems, electronic systems, and population dynamics [65, 66].

The following linear PDE is given as where , , , , , and . is the CFD of order , with the following initial condition:

The nonlinear PDE is as follows: with the initial condition

*Definition 1 (see [67]) (a novel fractional Taylor series formula in E-T). *Assume that is a piecewise continuous and exponential order and that the E-T of is provided by a fractional Taylor series.
where is the coefficient of the novel fractional Taylor series formula in E-T.

Lemma 2. *Assume that and are piecewise continuous and of exponential-order functions, and , , and are constants. Then, the following axioms hold [67]:
*(i)*(ii)**(iii)*

The framework of this study is as follows. The fundamental recommendation beyond the ERPSM with convergence and absolute error analysis for FMPS is demonstrated in Section 2. We also demonstrated numerical examples of FMPS to exemplify the competency, potential, and straightforwardness of the new combined technique. The linear and nonlinear PDEs are discussed in Sections 3 and 4, respectively. Finally, our findings are summarized in Section 5.

#### 2. The ERPSM to Demonstrate the FMPS

In this section, we use ERPSM to construct the solutions of the FMPS as in Equations (4) and (5). The main algorithm of this method for solving the FMPS can be summarized by the following steps: applies the E-T to Equation (4). As a result, we get an algebraic form in E-T space. In the second step, using the novel fractional Taylor’s series formula in E-T, we represent the solution in the E-T space of the algebraic equation obtained in the first step. The coefficients of this expansion are determined with the help of residual function and limit concept. As a result, we have found a solution to the problem in its original space by taking the inverse E-T.

In the next subsection, we derive the main algorithms of the ERPSM for the FMPS.

##### 2.1. The Algorithm of ERPSM for Solving FMPS

We use the following algorithm to create the solution with the help of ERPSM for the FMPS as shown in Equations (4) and (5):

*Step 1. *Rewrite Equations (4) and (5). We have

*Step 2. *We get the following result by implementing the E-T on both sides of Equations (12) and (13).

By utilizing the second part of Lemma 2, Equations (14) and (15) become as follows: where

*Step 3. *Assume that algebraic equations (16) and (17) have the solution in the expansion form as

The th-truncated series of and are as

*Step 4. *By utilizing the following lemma,

The th-truncated series becomes

*Step 5. *Consider the Elzaki residual function (ERF) of Equations (22) and (23) separately, as well as the th-truncated Elzaki residual functions (ERFs), so that

*Step 6. *Replace the succession arrangement of and in Equations (26) and (27).

*Step 7. *To highlight important facts, we extend some features that arise in the RPSM [67–69].

That is obvious.

Therefore,

Since, It is clear that As a result, we get the following:

*Step 8. *Replace the th attained values of and into the th-truncated series of and to become the th-approximate explanation of Equations (16) and (17).

*Step 9. *Use the inverse E-T on and to obtain the th-approximate solution and in the original space.

The next theorem clarifies and establishes the conditions for the series solutions to converge.

Theorem 3. *Let be a Banach space denoted with a suitable norm over which the sequence of partial sums is defined. Assume that the initial guess remains inside the ball of the solution . Then, the series solution converges if such that .*

*Proof. *A sequence of partial sums is defined as
Next is that we would have to show that is a Cauchy sequence in . To demonstrate this, consider the following relationship:
where .

For every , we have
From triangle inequality, we have
and
Therefore,
Showing that the sequence is bounded, we can obtain for that

This proves that the sequence of partial sums generated by ERPSM is Cauchy and hence convergent.

In the next theorem, we determine the maximum truncation error.

Theorem 4. *Let be the approximate solution of the truncated finite series Assume it is attainable to acquire a real number , in order that ; furthermore, the utmost absolute error is
*

*Proof. *Let the series be finite, then
This proof is complete.

In the next subsection, two problems of FMPS are established to illustrate the performance and appropriateness of the proposed method.

##### 2.2. Numerical Examples

To demonstrate the execution and capability of ERPSM, we investigated two interesting and important problems for FMPS:

*Problem 5. *Consider the following FMPS:
with the initial conditions and

Applying the E-T to Equations (40) and (41), we get

We have the following results from Equations (42) and (43) using the procedure mentioned in Subsection 2.1:

Assume that Equations (44) and (45) have a series solution in the following form:

The th-truncated expansion is as follows:

By using the first part of Lemma 2, the th-truncated series becomes

The ERFs are formulated as

The th-truncated ERF takes the following form:

Substitute into Equations (65) and (66) and solve the following expression to find the unknown coefficients.

Thus, we have

We have the following 5th-order approximate solution in the original space for Equations (40) and (41) when uses the procedure mentioned in Subsection 2.1:

Equations (53) and (54) represent the first five terms of and , respectively, and therefore the exact solutions of Equations (40) and (41).

The following 2-D graphs show the absolute and relative error for Example 1.

Figures 1 and 2 demonstrate the 2-D graphs of absolute and relative errors in the intervals over the ten-step approximate and exact solutions of Equation (40) at , respectively. According to the figures, the approximate solution is extremely close to the precise solution. Figures 3 and 4 are graphs of absolute and relative errors in the intervals over the ten-step approximate and exact solutions of Equation (41) at , respectively. One can perceive the equivalent verdicts depicted for Equation (40).

Error functions are presented to observe the exactness and capability of the numerical method. To prove the exactness and capability of ERPSM, we selected three kinds of error functions, such as absolute, residual, and error functions.

Table 1 shows the absolute and relative errors at reasonable nominated grid points in the interval among the five-step approximate and exact solutions of Equations (40) and (41) at attained using ERPSM. Table 1 shows that the approximate and exact solutions are quite close to each other, confirming the effectiveness of the recommended strategy.

*Problem 6. *Consider the following FMPS:
with the initial conditions

Applying the E-T to Equations (55) and (56), we get

Using the approach mentioned in Subsection 2.1, we obtain the following results from Equations (57) and (58) as

Assume that Equations (59) and (60) have the series solution in the following form:

The th-truncated expansions are as

By using the first part of Lemma 2, the th-truncated expansions become

The ERFs are formulated as

The th-truncated ERF takes the following form:

Substitute into Equations (65) and (66) and solve the following expression to find the unknown coefficients.

Thus, we have

We have the following 5th-order approximate solution in the original space for Equations (55) and (56) when using the procedure mentioned in Subsection 2.1:

Equations (69) and (70) represent the first five terms of and , respectively, and therefore the exact solutions of Equations (55) and (56).

In the next section, we will use our new ERPSM to find approximate and exact solutions to linear PDE.

#### 3. The ERPSM to Demonstrate the Linear PDE

In this section, we use our new ERPSM to construct the solutions of the linear PDE as in Equation (7). In the next subsection, we derive the main algorithms of the ERPSM for the linear PDE.

##### 3.1. The Algorithm of ERPSM for Solving Linear PDE

In this subsection, we exploit ERPSM to create the solution to the linear PDE. We start by taking E-T on Equation (7). We get the following:

By using the second part of Lemma 2, Equation (71) becomes

Assuming that algebraic equation (72) has the solution in the expansion form as

The th-truncated series of is as follows:

By using the first part of Lemma 2, we have the following:

The th-truncated series becomes

Now, define the ERF in the following form:

The th-ERF is as follows:

To highlight basic points, we generalize certain features that arise in the RPSM [67–69].

It is understandable that

Therefore,

Since, It is obvious that As a result, we get the following:

To determine the first undefined coefficient substitute in Equations (76) and (78). As a result, we obtain as following:

Using Equations (83), (84), and (85) in (82), we get the following:

Dividing by Equation (87), we get

By using to the Equation (88), we get