Abstract
The objective of this paper is to compute the new modified method of variational iteration and the Laplace Adomian decomposition method for the solution of nonlinear fractional partial differential equations. We execute a comparatively newfangled analytical mechanism that is denoted by the modified Laplace variational iteration method (MLVIM) and Laplace Adomian decomposition method (LADM). The effect of the numerical results indicates that the double approximation is handy to execute and reliable when applied. It is shown that numerical solutions are gained in the form of approximately series which are facilely computable.
1. Introduction
Fractional calculus has tremendous applications in applied science such as mechanical engineering, heat conduction, viscoelasticity, electrode-electrolyte polarization, nanotechnology, diffusion equations, and nearly every part of science and technology. One of the most important uses of fractional calculus is in fractional partial differential equations (PDEs) as most natural phenomena could be modeled by PDEs. Significant perception has been given to exact and convergent solutions of partial differential equations that embrace fractional system derivative because of its new framework and applications in assorted fields.
Contemporary improvements of fractional differential equations are supported by the earliest examples of applications in scientific fields, such Liouville’s example. Due to the usual occurrence of FPDEs in different developments of engineering and science, many leading researchers in the field developed the study in this direction by both theoretical and application means [1–4]. The nonlinear oscillation could be represented by fractional derivatives and the fluid dynamic traffic pattern with fractional derivatives [5, 6]; this leads to the use of FPDEs. Several of the main tools used in solving this type of equations are LADM, VIM, and homotopy. It should be noted that LADM effects have shown higher accuracy as compared to other methods in the literature [7]. LADM and the VIM are comparatively new approaches to fit an analytical approximation to linear and nonlinear problems [8–12]. ADM is also compared with LADM to construe the solution of FPDEs given in [13]. The investigation shows that the emergence of VIM and MVIM is strong and simple in calculation when solving the heat equation in the Dirichlet boundary [14]. The MVIM is an approximation numerical algorithm to resolve nonlinear and nonhomogeneous differential equations [15]. In [16], they proposed a combined form for solving nonlinear interval Volterra-Fredholm integral equations of the second kind based on the modifying Laplace Adomian decomposition method. They have been concerned with the solution of Caputo fractional integrodifferential equations by the modified variational iteration technique via the Laplace approach [17]. From the computational viewpoint, the MVIM is more efficient, convenient, and easy to use [18].
In recent times, there has been a wide concentration in the VIM for solving a large range of equations such as algebraic, differential, partial-differential, functional-delay, and integral-differential equations. The main feature of this technique is to build a correction functional using a general Lagrange multiplier selected in a suitable way that its adjustment solution is better concerning the initial assessment function. The target of this work is to extend the application of MLVIM to outfit convergent solutions for initial value problems of nonlinear partial fractional differential equations and to create a rapprochement with that obtained by LADM.
2. Definitions and Preliminaries
Definition 1. Let a real function which is said to be the space if the there exists a real number such that where [1, 2].
Definition 2. The Riemann-Liouville fractional integral operator of order of a function is known as [1–4]
Definition 3. The fractional derivative of is known as [1–3] Hence, we have the following properties:
Definition 4. The Laplace transform of the fractional derivative is known as where is the Laplace transform of [8].
3. Laplace Adomian Decomposition Method (LADM)
In this section, we will consider a class of nonlinear fractional partial differential equation of the form where meaning the fractional derivative is of the order where is a linear operator, is a nonlinear term function, and is the source function. The initial and boundary associated with equation (5) are of the from
Applying the Laplace transform to both sides of equation (5) by using linearity of Laplace transform, the result is
Now, using property of Laplace transform, we get
The standard Laplace decomposition method defines the solution known by the series
The nonlinear operator is decomposed as
is the mean Adomian polynomial [3, 4] that is given by
Substituting equations (10), (11), and (12) into equation (9), we have
Both sides of equation (13) yield the following iterative algorithm:
Applying inverse Laplace transform equations (14) and (15), we obtain
4. Modified Laplace Variational Iteration Method (MLVIM)
The new tactic of the modified Laplace variation iteration technique is instituted on the following steps.
Step 1. Removing the fractional derivative of order with respect to unknown function by applying Laplace and inverse Laplace transforms of equation (5), we obtain
Step 2. Differentiating the results obtained in Step 1 with respect to , we then get the value of the general Lagrange multiplier, for the correction functional iterative formula to equal one
Therefore, equation (18) can be put above in the following formula:
The general Lagrange multiplier for equation (19) can be identified optimally via variation theory to get
Substituting into equation (19), we get the iterative formula as follows:
Equation (21) is the call new modified function of Laplace transform and variational iteration method. Start with the initial iteration . The exact solution is present as the sequent approximation ; in other words, .
5. Illustrative Examples
In this section, we first implement the Laplace decomposition method and apply the fractional modified Laplace variational iteration method (MLVIM) for solving time-fractional partial differential equations in a tube.
Example 5. Assume the nonlinear time-fractional advection partial differential equation [19]. Follow the first condition
Now, by enforcement of the Laplace transform to the sides of equation (22), we get By using the initial condition, we obtain Taking the inverse Laplace transform to the sides of equation (25), we obtain
5.1. Laplace Adomian Decomposition
By applying the Laplace’s standard decomposition method to determine the solution known by the series equations (11) and (12) into equation (26), we obtain
The few ingredients of Adomian polynomials We have the following recurrent relations: The third term approximate solution for equation (22) is given by
5.2. Modified Laplace Variational Iteration
Now the new tactic of the modified Laplace variational iteration technique is instituted on Step 2 from equation (26), and we get
We simply substitute equation (31) and the initial condition equation (23) into equation (21); by the new modified function, we find
Example 6. Assume the following nonlinear fractional partial differential equation [20]: Follow the first condition,
(a)
(b)
(c)
(d)
Now, by enforcement of the Laplace transform to the sides of equation (33), we obtain By using the initial condition, we obtain
Taking the inverse Laplace transform to the sides of equation (36), we obtain
5.3. Laplace Adomian Decomposition
By applying the standard Laplace decomposition method which defines the solutionknown by the series equations (14) and (15) into equation (37), we obtain
The few ingredients of Adomian polynomials We have the following recurrent relations:
The fourth term’s approximate solution for equation (33) is given by
5.4. Modified Laplace Variational Iteration
Now the new tactic of the modified Laplace variational iteration technique is instituted on Step 2 from equation (37), and we get We simply substitute equation (42) and the initial condition equation (34) into equation (21); by the new correction function, we find
Example 7. Assume the following nonlinear fractional partial differential equation [20]: Follow the first condition,
(a)
(b)
(c)
(d)
Now, by enforcement of the Laplace transform to the sides of equation (44), we obtain By using the initial condition, we obtain Taking the inverse Laplace transform to the sides of equation (44), we obtain
5.5. Laplace Adomian Decomposition
By applying the standard Laplace decomposition method which defines the solutionknown by the series equations (14) and (15) into equation (48), we obtain
The few components of Adomian polynomials
We have the following recurrent relations:
The fourth term’s approximate solution for equation (44) is given by
5.6. Modified Laplace Variational Iteration
We simply substitute equation (53) and the initial condition equation (45) into equation (21); by the new correction function, we find
(a)
(b)
(c)
(d)
6. Conclusion
In this paper, we applied the modified Laplace variational iteration method (MLVIM) and Laplace Adomian decomposition methods (LADM) to obtain the exact solution of nonlinear fractional partial differential equations. We studied the convergence of LADM and MLVIM with the exact solution and the efficacy of the methods shown by the table, and we find that the convergence of LADM results better than MLVIM. The solutions obtained by the two methods are derived by infinite approximate series. Numerical examples are offered to clarify the performance and accuracy of the two methods.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.