Approximate Analytical Solution of Two-Dimensional Nonlinear Time-Fractional Damped Wave Equation in the Caputo Fractional Derivative Operator
In this work, we proposed a new method called Laplace–Padé–Caputo fractional reduced differential transform method (LPCFRDTM) for solving a two-dimensional nonlinear time-fractional damped wave equation subject to the appropriate initial conditions arising in various physical models. LPCFRDTM is the amalgamation of the Laplace transform method (LTM), Padé approximant, and the well-known reduced differential transform method (RDTM) in the Caputo fractional derivative senses. First, the solution to the problem is gained in the convergent power series form with the help of the Caputo fractional-reduced differential transform method. Then, the Laplace–Padé approximant is applied to enlarge the domain of convergence. The advantage of this method is that it solves equations simply and directly without requiring enormous amounts of computational work, perturbations, or linearization, and it expands the convergence domain, leading to the exact answer. To confirm the effectiveness, accuracy, and convergence of the proposed method, four test-modeling problems from mathematical physics nonlinear wave equations are considered. The findings and results showed that the proposed approach may be utilized to solve comparable wave equations with nonlinear damping and source components and to forecast and enrich the internal mechanism of nonlinearity in nonlinear dynamic events.
Linear and nonlinear fractional differential equations can successfully simulate fractional derivatives in a range of scientific and technical domains, including electrical networks, chemical physics, control theory of dynamical systems, reaction-diffusion, signal processing, and heat transform [1–7]. Because fractional differential equations (FDEs) often exist in several fields of engineering and science, many researchers focus their efforts on obtaining exact/approximate solutions to these dynamic fractional differential equations utilizing a variety of powerful established approaches, including the finite difference method , Caputo fractional-reduced differential transform method [9–11], Padé–Sumudu–Adomian decomposition method , triple Laplace transform method [13–15], double Sumudu transform iterative method , shifted Chebyshev polynomial-based method , Laplace decomposition method [18, 19], homotopy analysis method , double Laplace transform method , homotopy perturbation method [22, 23], conformable reduced differential transform method , conformable fractional-modified homotopy perturbation, Adomian decomposition method , differential transform method [26–28], and the new function method based on the Jacobi elliptic functions .
Among the approaches listed above, Keskin and Oturance were the first ones to present the Caputo FRDTM, which has been successfully utilized to solve linear and nonlinear fractional differential equations [9, 30]. Many intellectuals have implemented this method for solving various sorts of equations in recent years. For example, Kenea  used CFRDM to find closed solutions and accurate solutions to one-dimensional time-fractional diffusion equations with beginning conditions in the form of infinite fractional power series. CFRDTM offers the benefit of minimizing the number of computations required and offering analytic approximations, in many cases exact answers, in the form of a fast-converging power series with elegantly computed terms [32–35]. Furthermore, CFRDTM has an alternative plan to solve problems to overcome the drawbacks of well-known numerical and analytical methods such as Adomian decomposition, differential transform, homotopy perturbation, and variational iteration, which suffer from discretization, linearization, or perturbations [36–38].
The main purpose of this study is to introduce the LPCFRDTM, which is a new approach for solving the two-dimensional time-fractional nonlinear damped wave equation. The CFRDTM, the Laplace transform method, and the Padé approximant are all jointly used in this procedure. The Padé approximation has been used in a variety of domains to approximate rational series solutions; it was invented by Henri Padé  circa 1980. Baker  established the existence and convergence of subsequences. The Padé approximant method outperforms other series approximation methods and is used to manage series convergence. The authors of the paper  used the multivariate Padé approximation method for solving the European vanilla call option pricing problem. According to the relationships of “smaller than” or “greater than” between stock price and option exercise price, the Padé polynomials have appeared in the fractional Black-Scholes equation using the provided method. Using these polynomials, they applied the multivariate Padé approximation method and calculated numerical solutions of the fractional Black–Scholes equation for both situations. The obtained results reveal that the multivariate Padé approximation is a very quick and accurate method for the fractional Black–Scholes equation. The Padé approximants, in other words, heighten the domain of convergence of the truncated power series solution achieved via CFRDTM or other methods, leading to the exact solutions in many cases [12, 42–45].
The proposed LPCFRDTM technique has been utilized to solve the problems as follows: The CFRDTM is used to derive the solution to FDEs in convergent power series form. Second, even though the CFRDTM solution series has a high number of terms, it may converge in a narrow area. As a result, the LPCFRDTM magnifies the truncated power series’ convergence domain, typically resulting in the exact solution. We use the Laplace transform to improve the solution of convergent series generated by the CFRDTM and then form its Padé approximant to turn the transformed series into a meromorphic function. Finally, to achieve the approximate analytical solution, we use the inverse Laplace transform of the Padé approximant function. The capacity to widen the domain of convergence of solutions or include discovering exact answers is a major benefit of using this method. Also, the LPCFRDTM can obtain exact solutions without any perturbation parameters like HPM [27, 46, 47].
The generalized two-dimensional dynamical time-fractional nonlinear damped wave equation with a source term in the Caputo fractional derivative operator is taken into account in this article : where
The initial condition associated with equation (1) is given bywhere is the scalar variable, is the time, and the parameters are supposed to be real numbers with is the alleged dissipative term. When , Equation (1) reduces to the undamped wave equation, while to the damped one. The known functions and represent wave kinks or modes and velocity, respectively.
The remainder of this work is divided into the following sections. The fundamental definitions of fractional calculus are provided in Section 2. CFRDTM is introduced in Section 3 along with definitions and its convergence analysis in subsection 3.1. The main idea behind the Padé approximant is explained in Section 4. Section 5 explains the underlying premise of the Laplace–Padé resummation method. In Section 6, we demonstrate the proposed method’s reliability, convergence, and efficiency using four illustrative instances. Approximate analytical answers and numerical simulations are presented in tables and graphs in Sections 6.1 and 6.2, respectively. In Section 7, we have a quick discussion. Finally, a conclusion is formed in section 8.
Definition 1. The Riemann–Liouville fractional derivative operator of is given bywhere the gamma function is simply a generalization of the factorial real arguments and defined by
Definition 2. The Riemann–Liouville fractional integral of order for a function with , and is defined asWhen trying to describe real-world issues with fractional differential equations, the Riemann–Liouville derivative has several drawbacks because it necessitates the definition of fractional order beginning conditions, which have yet to be physically explained. In their work on the theory of viscoelasticity, Caputo and Mainardi  suggested a modified fractional differentiation operator to address this mismatch. With the Caputo fractional derivative, which has unambiguous physical implications, initial and boundary conditions involving integer-order derivatives can be employed.
Definition 3. From Caputo’s perspective, the fractional derivative of is defined asIn particular, if , then the Caputo fractional derivative of is
Lemma 1. If and then
3. Caputo Fractional-Reduced Differential Transform Method (CFRDTM)
In this section, the fundamental necessary concepts and operations of the dimensional CFRDTM [30, 32, 43] are presented. Additionally, the convergence analysis of the CFRDTM is also presented in subsection 3.1.
Definition 4. If a function is analytic and differentiated continuously with respect to the time variable and space variables in the domain of interest, then letwhere is the -dimensional spectrum function or the transformed function, and the parameter indicates the order of the time fractional derivative. The original function is represented by lowercase in this article, whereas the transformed function is represented by uppercase .
Remark 1. The function is represented in real life by an infinite series of Equation (9) about and can be expressed as where the tail function is negligibly small.
Moreover, the inverse CFRDT of the set of yields an approximation solution as follows:where is the approximate solution’s order. As a result, the actual answer to the problem is obtained as follows:From Equation (11), the principle of the CFRDTM can be determined to be derived from the power series expansion.
Theorem 1. (see ). Assume that , and are the RDT of the functions , and respectively, then we have the following equations:(i)If then(ii)If then
Remark 2. The Mittag–Leffler function, which is a generalization of the exponential function, is defined as For reduces to and
Theorem 2. ) Caputo fractional-reduced differential transform of the initial condition.
To validate the fundamental concepts of the CFRDTM, we consider the nonlinear damping wave equation (1) with the initial condition (2) by applying the features of CFRDTM listed in Table 1 and Theorem on both sides of problem (1).The CFRDT of each term in (17) is given as follows.Substituting equation (13) into equation (12), we may construct the following iteration formula:Solving for we obtainWhen we apply the CFRDT to condition (2) according to Theorem 2, we get Using Equation (15) into Equation (14) and solving the resulting system for , we get the following values:and so on.
Then, the inverse CFRDT of the set of values gives the approximate solution:In the equation, is the order of approximation solution. Therefore, the exact solution to the considered problem can be obtained as follows:
3.1. Convergence of the Method
4. Pade Approximate
In numerical mathematics, Padé approximation  is believed to be the best approximation of a function by rational functions of a given order. Under this technique, the approximate power series agrees with the power series of the function it is approximating. The Padé approximant often gives a better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge and also enlarges the domain of convergence of the truncated power series solution. For such reasons, Padé approximants are often used in many fields of computations.
Let be an analytical function with Maclaurin’s expansion.Then, the Padé approximant to of order , which is denoted as , is defined as follows [40, 42]:where we considered and the numerator and denominator have no common factors. The numerator and the denominator in equation (19) are built in such a way that , and their derivatives agree at up to That is,
By cross multiplying, we find that
Equating the coefficients of from (29), we find
Comparing the equal power of on both sides of (29), we get
Remark 3. For a fixed value of error equation (20) is the smallest when the numerator and denominator of Equation (14) have the same degree or when the numerator has one degree higher than the denominator.
5. Laplace–Pade Resummation Method
The LPCFRDTM, which is a combination of the CFRDTM and the Laplace–Padé resummation method, is described as follows:(i)By means of CFRDTM, we first find the series solution of the given equation that is similar to series (16)(ii)Second, power series (16) is transformed using the Laplace transformation(iii)Next, we replace by in the resulting equation(iv)The series produced from (3) is then transformed into a meromorphic function by constructing its Padé approximant of order , where and are chosen at random but should be smaller than the series’ order(v)After that, in the resulting equation, we substitute by (vi)As a final point, we obtain the precise or approximate answer by applying the inverse Laplace transformation
6. Illustrative Examples
We offer four examples to establish the validity and efficiency of the suggested strategy. These cases’ solutions are also compared with exact solutions.
6.1. Analytical Solution to Illustrative Examples
Example 1. Consider the following dimensional hyperbolic time-fractional telegraph equation in the region as follows:Under the initial condition, we getApplying the properties of CFRDT listed in Table 1 on both sides of Equation (24), we obtainwhere is the Caputo fractional-reduced transform of , and it is obtained by using Remark 2.
When the CFRDT is applied to initial condition (25) in view of Theorem 2, it producesWe get the following values for using Equation (27) into Equation (26), recursively.
For , using the values of and from (35) into Equation (26), we getfor , using the values of and in Equation (26), we obtainContinuing in the same manner for , we obtainThen, using the inverse CFRDT (9) as a definition, we get.In this situation, the truncated series is approximated using the Laplace–Padé approximant up to order of perturbation. That is,After that, we apply the Laplace transform to both sides of (40), and we getFor straightforwardness, we write in the place of in Equation (31), and we getNow, we convert transformed series (32) into a meromorphic function by forming its Padé approximant of with and We obtain
Using (23) and (22), we getSolving for and , we obtain
Then, we obtainTherefore, all Padé approximants of the eqaution with and giveNow, by changing into in the equation, we obtain in terms of as follows:Finally, on both sides of Equation (34), using the inverse Laplace transform, we have the exact solution as follows:Putting in Equation (35), we obtain
Example 2. Consider the following two-dimensional hyperbolic nonlinear time-fractional sine-Gordon equation on the domain Under the initial condition, we getApplying the properties of CFRDT listed in Table 1, Theorem 1, and Definition 4 on both sides of Equation (37), we obtainwhere and are transformed forms of nonlinear term and respectively. Applying the CFRDT to initial condition (38) in view of Theorem 2, we getSubstituting Equation (40) into Equation (39) and applying Theorem 1, Definition 6, and properties of CFRDT listed in Table 1, we obtain the following successive iterated values for
For , using the values of and from (53) into (52), we getAccording to Theorem (1),
Using Definition (5), we getConsequently, we obtainfor , using the values of and in (52), we getAccording to Theorem (1), we getUsing Definition (5), we getConsequently, we obtainContinuing in the same manner for , we obtainThen, by using inverse CRDT (9), we getThe Laplace–Padé approximant is used to approximate the truncated series up to order of perturbation in this case. That is,Then, on both sides of (63), we use the Laplace transform to get