A New Method for Solving Sequential Fractional Wave Equations

In this article, we focus on two classes of fractional wave equations in the context of the sequential Caputo derivative. For the frst class, we derive the closed-form solution in terms of generalized Mittag–Lefer functions. Subsequently, we consider a more general class of nonhomogeneous fractional wave equations. Due to the complexity of fnding exact solutions for these problems, we employ a numerical technique based on the operational matrix method to approximate the solution. We provide several theoretical and numerical examples to validate the efectiveness of this numerical approach. Te results demonstrate the accuracy and efciency of the proposed method.


Introduction
Fractional partial diferential equations (FPDEs) have received increasing attention over the last few decades due to their applications in various felds, including physics, chemistry, engineering, and economics. Sequential fractional derivatives are a type of fractional derivative that involves the application of the fractional diferentiation operator multiple times in sequence. In other words, the output of the frst fractional diferentiation operator becomes the input to the next fractional diferentiation operator. Te sequential fractional derivative provides a more accurate description of complex systems, such as biological systems, exhibiting nonlocal and non-Markovian behavior. It also captures long-term memory efects that are not captured by integer-order derivatives. Tis makes it useful for modeling systems with long-term dependencies. Moreover, it is more fexible than other kinds of derivatives because it allows for a continuous range of fractional values, enabling a more precise modeling of systems with complex behavior. Sequential fractional derivatives have been successfully used in a wide range of applications, including control theory, signal processing, and image processing. Tey have been studied extensively in recent years due to their potential applications in various areas of science and engineering. For example, they have been used to model anomalous difusion phenomena in complex systems such as porous media, fractal structures, and biological tissues. Tey have also been used in the analysis of viscoelastic materials and in the design of control systems.
Several numerical methods have been proposed for the approximation of sequential fractional derivatives. One common approach is to use the recursive formula to compute the derivatives in sequence. Another approach is to use the fractional integration operator to convert the sequential fractional derivative into a single fractional derivative, which can then be approximated using existing numerical methods for fractional derivatives. Overall, sequential fractional derivatives are a powerful tool for modeling and analyzing complex systems, and their study is an active area of research in the feld of fractional calculus.
In [1], Yang introduces the concept of sequential fractional derivatives and discusses their properties and applications. Te author provides some examples to illustrate the use of sequential fractional diferential equations in modeling complex systems. In [2], the authors study the solutions of sequential fractional diferential equations and provide numerical examples to illustrate their results. Tey also introduce a new approach for solving sequential fractional diferential equations using the fractional Laplace transform. Torres [3] introduces the concept of sequential fractional variational calculus, which is an extension of the standard fractional variational calculus to include sequential fractional derivatives. Te author discusses the properties of sequential fractional variational calculus and provides some examples to illustrate its applications.
Te operational matrix method is a popular numerical technique for solving FPDEs. In [4], the authors developed a new approach for solving two-dimensional FPDEs using a combination of the operational matrix method and the Galerkin method. Te proposed approach was applied to a range of problems, including the fractional difusion equation. Te numerical results demonstrated the efciency and accuracy of the method.
Another study by Yousef and Delavari [5] focused on the application of the operational matrix method to solve a class of nonlinear FPDEs. Te authors developed a new technique that combines the operational matrix method with the Newton-Raphson method to solve a variety of nonlinear FPDEs. In [6], the authors developed a new approach for solving time-fractional difusion equations using a combination of the operational matrix method and the Tau method. Te proposed approach was applied to several problems, including the time-fractional difusion equation with a space-dependent coefcient. Moreover, Hesameddini et al. [7] proposed a new numerical technique based on the operational matrix method to solve multiterm timefractional partial diferential equations. Te authors applied the proposed method to a variety of problems, including the multiterm time-fractional difusion equation and the multiterm time-fractional wave equation.
Finally, in [8], a numerical technique based on the operational matrix method was developed to solve the timefractional difusion equation in two dimensions. Te authors used the proposed method to solve several problems, including the time-fractional difusion equation with a spacedependent coefcient. In [9], the authors developed a new numerical technique based on the operational matrix method to solve the time-fractional advection-difusion equation. Te proposed method was shown to be accurate and efcient for solving this type of equation. In [10], the authors proposed a new approach based on the operational matrix method to solve the space-time fractional difusion equation. Te proposed method was compared with several other numerical methods, and the results showed that it is accurate and efcient. In [11], the authors developed a new numerical technique based on the operational matrix method to solve the multiterm time-fractional difusion equation. Te proposed method was shown to be accurate and efcient, outperforming several other numerical methods. In [12], the authors developed a new numerical technique based on the operational matrix method to solve the fractional telegraph equation. Te proposed method was shown to be accurate and efcient and was compared with several other numerical methods. In [13], the authors developed a new approach based on the operational matrix method to solve the fractional difusion equation with a space-dependent coefcient. Several researchers have investigated the applications of fractional derivatives and developed numerical methods for solving such applications. In [14], extended fractional dynamical systems are studied, while [15] focuses on multispace fractional Korteweg-de Vries Equations. Te chaotic Lorenz system is solved using wavelet polynomials in [16], and [17] addresses the solution of delay diferential equations. In [18,19], the collocation method and series method are employed to solve fractional initial value problems (IVPs), respectively. However, [20] presents an iterative method developed for solving such problems. Te operational matrix method is used to solve systems of fractional IVPs in [21]. Te existence and uniqueness of fractional problems are examined in [22].
In this article, we compare and contrast the sequential Caputo derivative and the Caputo derivative. We extend these concepts to solve a class of homogeneous fractional wave equations with constant coefcients. We derive the analytical solution for a specifc case of a homogeneous second-order fractional equation with constant coefcients. We provide evidence to support our fndings and present two practical illustrations of their application.
Additionally, we address a diferent category of nonhomogeneous fractional wave equations with variable coefcients. Since fnding exact solutions for these problems is challenging and sometimes not feasible, we employ a numerical technique for approximate solutions. Te operational matrix method serves as the foundation for this approach. We derive the operational matrices to integrate the equations into a unifed system. Tis method ofers advantages such as low equation setup costs and avoids the need for projection methods like Galerkin or collocation.
To validate the efectiveness of this approach, we provide two examples for experimental testing. Te results demonstrate a close match between the exact and approximation solutions, as observed in the graphs. Furthermore, the L 2 -errors for various values of t are nearly zero, indicating the accuracy of the method.
For solving fractional partial diferential equations, the operational matrix method has several advantages. Te operational matrix method is an easy-to-understand technique for solving diferential equations. It transforms the task of solving a diferential equation into solving a set of algebraic equations, which can be efciently and accurately solved using numerical methods. It is highly efcient as it avoids directly solving the diferential equation and simplifes the problem into solving a system of algebraic equations.
Moreover, the operational matrix method provides precise solutions for a wide range of problems. It is fexible and can be applied to various types of diferential equations, making it suitable for diverse applications. Additionally, the operational matrix method can be combined with other numerical methods to enhance its capabilities and address specifc requirements.
Furthermore, the operational matrix method has a strong theoretical foundation, with well-established principles and extensive research support. Tis solid theoretical basis makes the operational matrix method a popular choice in scientifc and engineering applications. Its simplicity, efciency, and accuracy contribute to its widespread use in various felds. All of the aforementioned advantages motivate us to study and investigate the solution to this problem.
Tis research paper is divided into six sections. In sections one and two, we provide a brief literature review of fractional wave equations and the operational matrix method. Ten, we present some defnitions and results necessary for this paper. In section three, we provide a detailed description of the exact solution for special types of fractional problems. We fnd the fundamental set of solutions in the closed form for a class of fractional wave equations. In section four, we derive the numerical technique used to solve a class of nonhomogeneous wave equations. Proofs of our results are given in sections three and four. Several examples are provided in section fve to demonstrate the efciency of the proposed method. Finally, we summarize the main fndings of this study and draw conclusions in section six.

Basic Definitions and Results
Now, let's begin with the defnition of the Caputo derivative and the fractional integral operator.
Defnition 1 (see [23]). Let L ∈ N and p > 0. Defne the spaces C p and C L p as (1) where Γ is the Gamma function, and the fractional integral operator is defned by where L − 1 < p < L and L ∈ N.
Te frst important rule is the power rule, which is given by Te composites of the Caputo derivative and fractional integral operator are given by the following relations: If (1/2) < q ≤ 1 and L ∈ N, the power rule implies that However, From equations (6) and (7), one can see that in general. In this case, we say that the Caputo derivative is not sequential. Te formal defnition of the sequential Caputo derivative is given as follows.
Defnition 2 (see [24]). If L − 1 < Lp < L and for L ∈ 2, 3, . . . { }, then the Caputo derivative is called the sequential Caputo derivative of order p. In this case, we use the following notation sc D Lp w(z).
Tis property is an important property, and it will afect the solution of the fractional wave equations. Before we go further in this paper, we need the following result.
) be the Mittag-Lefer function. If 0 < p < 1 and μ be any constant, then while □ For more details about fractional derivatives and their properties, we refer the reader to the following references. In [23], the properties of fractional derivatives and some of their applications are discussed. Additionally, Podlubny [25] provides a theoretical discussion on fractional calculus. In [24], the authors compare sequential and nonsequential fractional derivatives. From now on, we use the following notation D p to mean sc D p .
We study the exact and numerical solutions of the following class of fractional wave equations in this article with ](x, 0) � y(x), where α(t) and β(t) are continuous functions in [0, ∞),

Homogeneous Fractional Wave Equation with Constant Coefficients
In this section, we investigate the exact solution of the following special case of problems (13)-(14) with where α and β are real constants. Now, we start with the frst result which is given by the following theorem.
Proof. Using the fact Γ(z + 1) � zΓ(z) and equation (4), we get Ten, which can be written as

Journal of Mathematics
For j � 1, 2, . . ., the coefcient of t jp is simplifed as Since μ 2 + αμ + β � 0 and μ � (−α/2). For j � 0, we have Terefore, which completes the proof. □ Theorem 5. Let α ≠ 0 and β be two constants and (1/2) < p ≤ 1. Ten, the exact solution of the following fractional diferential equation: is given by where Proof. We divided the proof into two cases. In the frst case, we assume that α 2 ≠ 4β. Ten, simple calculations imply that which yield using equation (11) to the following equations: Tus, E p (μ 1 t p ) and E p (μ 2 t p ) are solutions to equation (26). Now, we want to show that they are linearly independent. To show that, we set c 1 E p (μ 1 t p ) + c 2 E p (μ 2 t p ) � 0. Ten, Journal of Mathematics 5 which implies that For j � 0 and 1, we get 1 1 Since det 1 1 then c 1 � c 2 � 0. Tus, E p (μ 1 t p ) and E p (μ 2 t p ) are linearly independent. Now, we consider the case when α 2 � 4β. Ten, using Teorem 4, we have Ten, E p ((−α/2)t p ) and t p E p,p ((−α/2)t p ) are solutions to the problem (26). Now, let When t � 0, direct substitution in equation (36) implies that c 1 � 0. When t � 1, we will get c 2 E p,p (−α/2) � 0 which implies that c 2 � 0. Tis completes the proof of the theorem. □ Now, we discuss the solution of problems (16)-(17) in the following theorem. Theorem 6. Let α ≠ 0 and β be two constants, (1/2) < p ≤ 1. Ten, the exact solution of the following fractional problem with is given as where Proof. We use the separation of variables technique. Let

Ten, substitute equation (39) into equation (37) to get
Teorem 5 implies that the solution of equation (43) is given as Terefore, the solution of problems (37)- (38) is If we have initial conditions, we can use the Fourier series expansion to fnd a j and b j for j ∈ N.

Nonhomogeneous Fractional Wave Equation
In this section, we introduce a numerical technique for solving problems (13)- (15) with α(t) � 1 and κ(x) � 1. Te proposed numerical approach is based on the operational matrix method (OMM). Since the functions involved are two-dimensional, we defne the two-dimensional block pulse functions (BPFs).

Theorem 9. Te operational matrix of the Riemann integral
Proof. For any s ∈ 0, 1, . . . , L 1 − 1 , simple calculations imply that Terefore, implies that which gives the result of the theorem.

□
Note that I x Φ * (x) � Φ * (x)R * I . Using equations (62) and (66) and by taking the Riemann integral, we get Let ] x (0, t) � q(t). Ten, we can rewrite it as Since then where Q � Ten, Using the condition ](0, t) � 0 and by taking the Riemann integral one more time, we get

Journal of Mathematics
Since φ j (x) Now, we study the solution of problems (13)-(15) using the OMM numerically.
Example 13. Consider the following problem: where Ten, the exact solution of problems (111)- (113) is Let the L 2 -error with respect to x is defned by Ten, the error for diferent values of t is reported in Table 1. In this example, we use L 1 � L 2 � 40. Te graph of the exact and approximate solutions for t � 0.3j, j � 0, 1, . . . , 5 is given in Figure 1. Note that the exact solution in Figure 1 is given as a solid line while the approximate solution is as dots. Also, the graph of the exact and approximate solutions on the domain [0, 1] × [0, 1.5] is given in Figure 2.
Example 14. Consider the following problem: where Ten, the exact solution of problems (111)-(113) is Ten, the error for diferent values of t is reported in Table 2. In this example, we use L 1 � L 2 � 45. Te graph of the exact and approximate solutions for t � 0.3j, j � 0, 1, . . . , 5 is given in Figure 3. Note that the exact solution in Figure 3 is given as a solid line while the approximate solution is given as dots. Also, the graph of the exact and approximate solutions on the domain [0, 1] × [0, 1.5] is given in Figure 4. Now, we want to compare our results with [26]. where Ten, the exact solution of problems (111)- (113) is        We will compare our results with [26] based on the maximum error for diferent values of p. For p � 0.75, we take ∆x � (1/45), � (1/220), and for p � 0.0.85, we take ∆x � (1/70), � (1/480) for the results in [26]. Tese results are reported in Table 3.

Conclusions
In this article, we discuss the diferences between the sequential Caputo derivative and the Caputo derivative. We generalize this concept to fnd the solution of a class of homogeneous fractional wave equations with constant coefcients. Specifcally, we derive the analytical solution of a homogeneous second-order fractional equation with constant coefcients in one variable. We provide evidence to support our fndings and illustrate their practical application through two examples. We then shift our focus to a diferent category of variable coefcient nonhomogeneous fractional wave equations. Since fnding exact solutions for these problems can be challenging or even impossible, we employ a numerical technique based on the operational matrix method (OMM) to approximate the solutions. Te OMM allows us to reduce the fractional order diferential equations to algebraic systems, ofering benefts such as low equation setup costs and the absence of projection methods like Galerkin or collocation.
To validate our approach, we provide two examples and compare the exact and approximate solutions. Te results demonstrate the accuracy and efectiveness of our strategy, as the graphs of the solutions closely match and the L 2 -errors are nearly zero for various values of t.
From these examples, we note the following observations: Example 11 presents the exact solutions of the homogeneous second-order fractional equation with constant coefcients for diferent choices of α and β based on Teorem 5. Example 12 provides the exact solution of the homogeneous fractional wave equation with constant coefcients based on Teorem 6. Examples 13 and 14 demonstrate the small L 2 -errors within O(10 − 15 ), as shown in Tables 1 and 2, respectively. Te approximate solution converges to the exact solution, as depicted in Figures 1 and 4. Figures 1 and 3 display the exact and approximate solutions for diferent values of t, illustrating their coincidence. Te operational matrix method (OMM) proves to be an efcient tool for solving such problems and can be generalized to other problems in physics and engineering. For future work, we plan to extend our study to investigate other types of problems, including difusion and Laplacian equations. Additionally, we aim to utilize the sequential Caputo derivative to solve systems of fractional differential equations with constant coefcients.

Data Availability
No data were used to support this study.

Conflicts of Interest
Te authors declare that they have no conficts of interest.