Residual Power Series Technique Research Articles

Explicit and approximate series solutions for nonlinear fractional wave-like differential equations with variable coefficients

This work investigates the exact and accurate approximate series solutions of the strongly nonlinear wave-like differential equations of fractional order with variable coefficients utilizing the Laplace residual power series technique in the 2D, and 3D-space. The posed model is a notable model in mathematical physics, engineering physics, and biophysics, to characterize the variations in laser light intensity, and fluid particle velocity distributions in turbulent flows. In the meaning of Caputo fractional derivative, the posed models are solved analytically via simulation of the generalized Taylor formula in the Laplace space. Different from the other analytic methods, an easy recurrence formula for finding the Laplace power series coefficients is derived. To validate the proposed approach, it is applied to three interesting problems. The approximate analytical solutions of studied fractional problems are expressed in fast-converging Caputo-fractional Maclaurin formulas. The results are numerically and graphically analyzed to demonstrate the efficiency and applicability of the technique, as well as to examine the effects of partial arrangement on the behavior of the solutions. Ultimately, the results and numerical simulations indicate that the used technique is an effective approach and accurate tool for solving fractional evolution models and other fractional models arising in nonlinear phenomena.

Constructing the fractional series solutions for time-fractional K-dV equation using Laplace residual power series technique

Constructing the fractional series solutions for time-fractional K-dV equation using Laplace residual power series technique

Analytical Approximate Solutions of Caputo Fractional KdV-Burgers Equations Using Laplace Residual Power Series Technique

The KdV-Burgers equation is one of the most important partial differential equations, established by Korteweg and de Vries to describe the behavior of nonlinear waves and many physical phenomena. In this paper, we reformulate this problem in the sense of Caputo fractional derivative, whose physical meanings, in this case, are very evident by describing the whole time domain of physical processing. The main aim of this work is to present the analytical approximate series for the nonlinear Caputo fractional KdV-Burgers equation by applying the Laplace residual power series method. The main tools of this method are the Laplace transform, Laurent series, and residual function. Moreover, four attractive and satisfying applications are given and solved to elucidate the mechanism of our proposed method. The analytical approximate series solution via this sweet technique shows excellent agreement with the solution obtained from other methods in simple and understandable steps. Finally, graphical and numerical comparison results at different values of α are provided with residual and relative errors to illustrate the behaviors of the approximate results and the effectiveness of the proposed method.

A new approach in handling one-dimensional time-fractional Schrödinger equations

<abstract> <p>Our aim of this paper was to present the accurate analytical approximate series solutions to the time-fractional Schrödinger equations via the Caputo fractional operator using the Laplace residual power series technique. Furthermore, three important and interesting applications were given, tested, and compared with four well-known methods (Adomian decomposition, homotopy perturbation, homotopy analysis, and variational iteration methods) to show that the proposed technique was simple, accurate, efficient, and applicable. When there was a pattern between the terms of the series, we could obtain the exact solutions; otherwise, we provided the approximate series solutions. Finally, graphical results were presented and analyzed. Mathematica software was used to calculate numerical and symbolic quantities.</p> </abstract>

Complex solutions for nonlinear fractional partial differential equations via the fractional conformable residual power series technique and modified auxiliary equation method

The fractional-order nonlinear Gardner and Cahn–Hilliard equations are often used to model ultra-short burst beams of light, complex fields of optics, photonic transmission systems, ions, and other fields of mathematical physics and engineering. This study has two main objectives. First, the main objective of this investigation is to solve the fractional-order nonlinear Gardner and Cahn–Hilliard equations by using the modified auxiliary equation method, which is not found in the literature. Second, the exact and approximate solutions of these equations are obtained by utilizing the fractional conformable residual power series algorithm and the modified auxiliary equation method. For the analytical and numerical solutions to two equations, we employ two separate techniques and establish consistency between the precise answers that are derived and the compatible numerical solution. To the best of our knowledge, this method of solving equations has never been investigated in this manner. The 2D and 3D contours have been defined using appropriate parametric values to support the physical compatibility of the results. The assessed findings suggested that the approach used in this study to recover inclusive and standard solutions is approachable, efficient, and faster in computing and can be considered a useful tool in resolving more complex phenomena that arise in the field of engineering, mathematical physics, and optical fiber.

Anew Technique to Obtain Analytical Solutions for The Fractional Foam Drainage Formula

In the present study, an innovative approach is proposed to solve a special case of fractional foam drainage using the Laplace residual power series technique (LRPS) in conjunction with the Caputo operator for determining the fractional derivative. The study provides extensive guidelines for utilizing this approach to solve time-fractional nonlinear formulas. The effectiveness and validity of the proposed method are investigated and established by comparing the obtained results with the accurate responses using graphs. The study also confirms that the accuracy of the proposed technique increases with the number of items in the combined solution of the problems, as demonstrated by the convergence of the correlation between the obtained solutions and the actual solutions for the special case of fractional foam drainage formula. The research findings suggest that the proposed technique is not only accurate and uncomplicated but also highly adaptable, making it suitable for addressing both linear and nonlinear situations.

Computational Study for Fiber Bragg Gratings with Dispersive Reflectivity Using Fractional Derivative

In this paper, the new representations of optical wave solutions to fiber Bragg gratings with cubic–quartic dispersive reflectivity having the Kerr law of nonlinear refractive index structure are retrieved with high accuracy. The residual power series technique is used to derive power series solutions to this model. The fractional derivative is taken in Caputo’s sense. The residual power series technique (RPST) provides the approximate solutions in truncated series form for specified initial conditions. By using three test applications, the efficiency and validity of the employed technique are demonstrated. By considering the suitable values of parameters, the power series solutions are illustrated by sketching 2D, 3D, and contour profiles. The analysis of the obtained results reveals that the RPST is a significant addition to exploring the dynamics of sustainable and smooth optical wave propagation across long distances through optical fibers.

Analytical analysis of fractional nonlinear Jaulent-Miodek system with energy-dependent Schrödinger potential

In this work, a novel technique is considered for analyzing the fractional-order Jaulent-Miodek system. The suggested approach is based on the use of the residual power series technique in conjunction with the Laplace transform and Caputo operator to solve the system of equations. The Caputo derivative is applied to express the fractional operator, which is more suitable for modeling real-world phenomena with memory effects. As a real example, the proposed technique is implemented for analyzing the Jaulent-Miodek equation under suitable initial conditions. Additionally, the proposed technique’s validity (accuracy and effectiveness) is examined by studying some numerical examples. The obtained solutions show that the suggested technique can provide a reliable solution for the fractional-order Jaulent-Miodek system, making it a helpful tool for researchers in different areas, including engineering, physics, and mathematics. We also analyze the absolute error between the derived approximations and the analytical solutions to check the validation and accuracy of the obtained approximations. Many researchers can benefit from both the obtained approximations and the suggested method in analyzing many complicated nonlinear systems in plasma physics and nonlinear optics, and many others.

Effective transform-expansions algorithm for solving non-linear fractional multi-pantograph system

<abstract> <p>This study presents a new and attractive analytical approach to treat systems with fractional multi-pantograph equations. We introduce the solution as a rapidly-converging series using the Laplace residual power series technique. This method controls the range of convergence and can be easily programmed to find many terms of the series coefficients by computer software. To show the efficiency and strength of the proposed method, we compare the results obtained in this study with those of the Homotopy analysis method and the residual power series technique. Furthermore, two exciting applications of fractional non-homogeneous pantograph systems are discussed in detail and solved numerically. We also present graphical simulations and analyses of the obtained results. Finally, we conclude that the obtained approximate solutions are very close to the exact solutions with a slight difference.</p> </abstract>

A hybrid analytical technique for solving multi-dimensional time-fractional Navier-Stokes system

<abstract> <p>In this research, a hybrid method, entitled the Laplace Residual Power Series technique, is adapted to find series solutions to a time-fractional model of Navier-Stokes equations in the sense of Caputo derivative. We employ the proposed method to construct analytical solutions to the target problem using the idea of the Laplace transform and the residual function with the concept of limit at infinity. A simple modification of the suggested method is presented to deal easily with the nonlinear terms constructed on the properties of the power series. Three interesting examples are solved and compared with the exact solutions to test the reliability, simplicity, and capacity of the presented method of solving systems of fractional partial differential equations. The results indicate that the used technique is a simple approach for solving nonlinear fractional differential equations since it depends only on the residual functions and the concept of the limit at infinity without needing differentiation or other complex computations.</p> </abstract>

Analytical Solutions of the Nonlinear Time-Fractional Coupled Boussinesq-Burger Equations Using Laplace Residual Power Series Technique

In this paper, we present the series solutions of the nonlinear time-fractional coupled Boussinesq-Burger equations (T-FCB-BEs) using Laplace-residual power series (L-RPS) technique in the sense of Caputo fractional derivative (C-FD). To assert the efficiency, simplicity, performance, and reliability of our proposed method, an attractive and interesting numerical example is tested analytically and graphically. In addition, our obtained results show that this algorithm is compatible and accurate for investigating the fractional-order solutions of engineering and physical applications. Finally, Mathematica software 14 is applied to compute the numerical and graphical results.

Application of ARA-Residual Power Series Method in Solving Systems of Fractional Differential Equations

In this research, systems of linear and nonlinear differential equations of fractional order are solved analytically using the novel interesting method: ARA- Residual Power Series (ARA-RPS) technique. This approach technique is based on the combination of the residual power series scheme with the ARA transform to establish analytical approximate solutions in a fast convergent series representation using the concept of the limit. The proposed method needs less time and effort compared with the residual power series technique. To prove the simplicity, applicability, and reliability of the presented method, three numerical examples are proposed and simulated. The obtained results show that the ARA-RPS technique is applicable, simple, and effective to get solutions to linear and nonlinear engineering and physical problems.

Fractional Series Solution Construction for Nonlinear Fractional Reaction-Diffusion Brusselator Model Utilizing Laplace Residual Power Series

This article investigates different nonlinear systems of fractional partial differential equations analytically using an attractive modified method known as the Laplace residual power series technique. Based on a combination of the Laplace transformation and the residual power series technique, we achieve analytic and approximation results in rapid convergent series form by employing the notion of the limit, with less time and effort than the residual power series method. Three challenges are evaluated and simulated to validate the suggested method’s practicability, efficiency, and simplicity. The analysis of the acquired findings demonstrates that the method mentioned above is simple, accurate, and appropriate for investigating the solutions to nonlinear applied sciences models.

A Reliable Way to Deal with Fractional-Order Equations That Describe the Unsteady Flow of a Polytropic Gas

In this paper, fractional-order system gas dynamics equations are solved analytically using an appealing novel method known as the Laplace residual power series technique, which is based on the coupling of the residual power series approach with the Laplace transform operator to develop analytical and approximate solutions in quick convergent series types by utilizing the idea of the limit with less effort and time than the residual power series method. The given model is tested and simulated to confirm the proposed technique’s simplicity, performance, and viability. The results show that the above-mentioned technique is simple, reliable, and appropriate for investigating nonlinear engineering and physical problems.

Series Representations for Uncertain Fractional IVPs in the Fuzzy Conformable Fractional Sense.

Fuzzy differential equations provide a crucial tool for modeling numerous phenomena and uncertainties that potentially arise in various applications across physics, applied sciences and engineering. Reliable and effective analytical methods are necessary to obtain the required solutions, as it is very difficult to obtain accurate solutions for certain fuzzy differential equations. In this paper, certain fuzzy approximate solutions are constructed and analyzed by means of a residual power series (RPS) technique involving some class of fuzzy fractional differential equations. The considered methodology for finding the fuzzy solutions relies on converting the target equations into two fractional crisp systems in terms of ρ-cut representations. The residual power series therefore gives solutions for the converted systems by combining fractional residual functions and fractional Taylor expansions to obtain values of the coefficients of the fractional power series. To validate the efficiency and the applicability of our proposed approach we derive solutions of the fuzzy fractional initial value problem by testing two attractive applications. The compatibility of the behavior of the solutions is determined via some graphical and numerical analysis of the proposed results. Moreover, the comparative results point out that the proposed method is more accurate compared to the other existing methods. Finally, the results attained in this article emphasize that the residual power series technique is easy, efficient, and fast for predicting solutions of the uncertain models arising in real physical phenomena.

Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations

In this article, an attractive numeric–analytic algorithm, called the fractional residual power series algorithm, is implemented for predicting the approximate solutions for a certain class of fractional systems of partial differential equations in terms of Caputo fractional differentiability. The solution methodology combines the residual function and the fractional Taylor’s formula. In this context, the proposed algorithm provides the unknown coefficients of the expansion series for the governed system by a straightforward pattern as well as it presents the solutions in a systematic manner without including any restrictive conditions. To enhance the theoretical framework, some numerical examples are tested and discussed to detect the simplicity, performance, and applicability of the proposed algorithm. Numerical simulations and graphical plots are provided to check the impact of the fractional order on the geometric behavior of the fractional residual power series solutions. Moreover, the efficiency of this algorithm is discussed by comparing the obtained results with other existing methods such as Laplace Adomian decomposition and Iterative methods. Simulation of the results shows that the fractional residual power series technique is an accurate and very attractive tool to obtain the solutions for nonlinear fractional partial differential equations that occur in applied mathematics, physics, and engineering.

Construction of analytical solution for Hirota–Satsuma coupled KdV equation according to time via new approach: Residual power series

In this work, a modern and novel approach method called the residual power series technique has been applied to find an analytical solution for an important equation in optical fibers called the Hirota–Satsuma coupled KdV equation with time as a series solution. Comparison of the analytical approximate solution with the exact solution concluded that the present method is an important addition for analyzing a system of partial differential equations that have a strong nonlinear term. We also represented graphically and discussed the effect of initial condition parameters and reaction of time on the model.

Application of Laplace residual power series method for approximate solutions of fractional IVP’s

In this study, different systems of linear and non-linear fractional initial value problems are solved analytically utilizing an attractive novel technique so-called the Laplace residual power series approach, and which is based on the coupling of the residual power series approach with the Laplace transform operator to generate analytical and approximate solutions in fast convergent series forms by using the concept of the limit with less time and effort compared with the residual power series technique. To confirm the simplicity, performance, and viability of the proposed technique, three problems are tested and simulated. Analysis of the obtained results reveals that the aforesaid technique is straightforward, accurate, and suitable to investigate the solutions of the non-linear physical and engineering problems.

Promoted residual power series technique with Laplace transform to solve some time-fractional problems arising in physics

Physical applications involving time-fractional derivatives are reflecting some memory characteristics. These inherited memories have been identified as a homotopy mapping of the fractional-solution into the integer-solution preserving its physical shapes. The aim of the current work is threefold. First, we present a new technique which is constructed by combining the Laplace transform tool with the residual power series method. Precisely, we provide the details of implementing the proposed method to treat time-fractional nonlinear problems. Second, we test the validity and the efficiency of the method on the temporal-fractional Newell–Whitehead–Segel model. Then, we implement this new methodology to study the temporal-fractional (1+1)-dimensional Burger’s equation and the Drinfeld–Sokolov–Wilson system. Further, for accuracy and reliability purposes, we compare our findings with other methods being used in the literature. Finally, we provide 2-D and 3-D graphical plots to support the impact of the fractional derivative acting on the behavior of the obtained profile solutions to the suggested models.

Smooth expansion to solve high-order linear conformable fractional PDEs via residual power series method: Applications to physical and engineering equations

We present a fractional series solution (FSS) for a class of higher-order linear fractional PDEs. The fractional derivative in this class is considered in the conformable fractional derivative (Co-FD) sense. An appropriate expansion was introduced to reach an FSS that is consistent with the target equations in this research. The residual power series technique is used to determine the coefficients of the FSS. Five applications are tested to verify the effectiveness of the used method, as well as to compare the current results with the previous results for the same applications in which the fractional derivative was considered in the Caputo sense. Numerical and graphical comparisons are made to determine the compatibility of the behavior of the solution in the case of the use of the concept of Co-FD as a suitable alternative to the use of the concept of Caputo fractional derivative (Ca-FD) in the modeling of natural phenomena.