Abstract

In this paper, we introduce the Yang transform homotopy perturbation method (YTHPM), which is a novel method. We provide formulae for the Yang transform of Caputo-Fabrizio fractional order derivatives. We derive an algorithm for the solution of Caputo-Fabrizio (CF) fractional order partial differential equation in series form and show its convergence to the exact solution. To demonstrate the novel approach, we include some examples with detailed solutions. We use tables and graphs to compare the exact and approximate solutions.

1. Introduction and Motivation

Noninteger calculus is a popular field that is aimed at explaining real-world phenomena that are modeled with operators of fractional order. It is also a field that uses noninteger order derivatives for differentiation and integration. A fractional derivative is a sort of derivative with a noninteger order that meets specific conditions: we get the primary function when the order is zero, and we get the ordinary derivative when the order is one [1]. The memory effect and conserved illustrative physical properties are two advantages of fractional derivatives. Over time, more accurate and up-to-date studies have been revealed using these types of operators. In this sense, the theory of fractional calculus and its applications are gaining popularity around the world. Fractional order models incorporate all previous knowledge from the past due to the memory effect, making it better for them to forecast and analyze dynamical models more efficiently. Fractional order calculus has a wide range of applications in a variety of areas due to its efficient properties, including biology and physics. [2–4], economics and finance [5, 6], science and engineering [7, 8], and mathematical modeling and mechanics [9–11]. Since the kernel of Caputo and Riemann derivatives is singular, they have a problem with this kernel. Since the kernel is used to explain the physical system’s memory effect, it is clear that both derivatives cannot accurately interpret the memory’s full effect due to this limitation. Caputo and Fabrizio (CF) [12] suggested a new fractional operator with an exponential kernel in a recent attempt around the middle of the last decade. This derivative’s kernel is nonsingular, so the results are more reasonable than the classical one. We include some applications of the CF operator in [13–15].

There are two types of nonlinear equations: linear and nonlinear. Partially differential equations of fractional order are notoriously difficult to solve, and finding an exact solution is even more difficult. In applied mathematics, exact and numerical solutions to this type of equation are necessary. As a result, new techniques for obtaining analytical solutions that are relatively close to the exact solutions have been developed. Integral transformations were often used to solve the differential equations. Integral transforms are useful for solving IVPs and BVPs in differential and integral equations. Several authors implemented various kinds of integral transforms and examined their implications in different types of differential equations. The Laplace transform is the integral transform that is used the most [16]. In 1998, Watugala [17] was successfully introducing Sumudu transform to solve differential equations and control engineering problems. Recently, in 2011, T. Elzaki and S. Elzaki introduced new integral transform “Elzaki Transform” and used heavily in solving partial differential equations [18]. Also in 2013, Aboodh introduces “Aboodh Transform” and applies for solving partial differential equations [19]. There are many transform which are available in literature.

He formulated the homotopy perturbation method (HPM) [20] in 1999, which is a combination of the homotopy method and the classical perturbation technique and has been widely applied on both linear and nonlinear problems [21–23]. The importance of HPM is that it does not need a small parameter in the equation, so it reduces the drawbacks of traditional perturbation methods. The main purpose of this article is to apply newly introduced integral transform called “Yang Transform” discovered by Yang [24] with HPM to solve nonlinear fractional order PDEs. We solve two popular nonlinear PDEs through the proposed method. We obtain a power series solution in the context of a quickly convergent series, and just several iterations are needed to achieve very efficient results. There is no requirement for a method like discretizing the problem and no linearization for the nonlinear problem, and just a few iterations can lead to a solution that can be easily estimated using these techniques.

2. Preliminaries

We give the basic definitions which are needed in the rest of paper. For the sake of simplicity, we write the exponential-decay kernel as.

Definition 1 [12]. If , then the Caputo-Fabrizio (CF) derivative may be expressed as follows: is the normalization function with . However, if , then the above derivative is defined as follows:

Definition 2 [12]. The CF fractional integral may be expressed as follows:

Definition 3 [7]. For , the following result represents the Laplace transform of CF derivative:

Definition 4. [24]. The Yang transform of is defined as follows: where is transform variable and for some the integral on the right exists.

Remark 5. Yang transform of some useful functions is given below.

3. Main Work

First, we derive formula for Yang transform of Caputo-Fabrizio fractional derivative through the Yang-Laplace duality property. At the end of this section, we give some examples with detailed solution to check the validity and efficiency of the novel method.

Lemma 6 (Laplace-Yang duality). Let the Laplace transform of is , then .

Proof. From Equation (5), we can obtain another form of the Yang transform by substituting as Since , this implies that Put in (8), we have Thus, from Equation (7), we obtain Also from Equations (5) and (8), we obtain The relations (10) and (11) represent the duality relation between the Yang and Laplace transform.☐

Lemma 7. Let be a continuous function; then, Yang transform of CF derivative of is given by

Proof. The Laplace transform of the CF fractional is given by Also, we have that the relation between Laplace and Yang property, i.e., . To obtain the required result, we replace by in Equation (13), and we get The proof is completed.☐

4. Algorithm of the Proposed Method

In this part, we discuss the algorithms of fractional order differential equations involving exponential-decay kernel. We provide some examples with detailed solution and its comparison with exact solutions.

4.1. Application to Caputo-Fabrizio Fractional Differential Equations

First, we develop the solution procedure of general nonlinear Caputo-Fabrizio (CF) fractional partial differential equations through YTHPM. Let us take a general nonlinear CF PDE with nonlinear term and linear term as where the term represents the source term. Implement Yang transform to Equation (15), and one can achieve Applying inverse of Yang transform, we achieve where the term represents the source term and the given I.C (initial condition). Now, we utilize HPM:

We decompose the nonlinear term as where represents the He’s polynomial and is calculated through the formula:

Putting Equations (18) and (19) in Equation (17), we achieve

We achieve the following terms by comparing coefficients of ρ in (21):

Thus, we may write the acquired solution of Equation (15) as follows:

4.2. Convergence and Error Analysis

The following theorems are based on the method’s mechanism and address the original problem’s (15) convergence and error analysis.

Theorem 8. Let be the exact solution of (15) and let and , where denotes the Hilbert space. Then, the obtained solution will converge if , i.e., for any , such that , .

Proof. We make a sequence of . To get the desired result, we have to prove that forms a “Cauchy sequence.” Further, let us take For , we acquire Since , and is bounded, let us take , and we obtain Thus, forms a “Cauchy sequence” in . It follows that the sequence is a convergent sequence with the limit for . Hence, this ends the proof.☐

Theorem 9. Let is finite and represents the obtained series solution. Let such that , then the following relation gives the maximum absolute error.

Proof. Since is finite, this implies that . Consider This ends the theorem’s proof.☐

4.3. Test Problems

The Yang transform homotopy perturbation method is applied to well-known nonlinear fractional PDEs in this section, demonstrating its ease of use and high accuracy. The space where the solution of the following examples lies is the Hilbert space .

Example 1. We take nonlinear KdV equation as follows: subjected to I.C .

Solution 1. Implementing the Yang transform to Equation (30), we have Applying Yang inverse transform, we have The solution via HPT is as follows: Thus, Equation (32) can be written as where is He’s polynomial which represents the nonlinear term . The first three terms can be written as Comparing the like powers of , we obtain Now, using He’s polynomials, we get . The second approximation is given by The third approximation is given by The second term of He’s polynomial is calculated as After simple calculation, we get Now, substituting Equation (40) into Equation (38), we get Similarly, one can compute the other terms. Thus, the approximate solution is given.