Residual Power Series Technique Research Articles

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.

A New Algorithm of Residual Power Series (RPS) Technique

Approximate analytic solutions of time–space fractional heat and wave equations are described. A new algorithm of Residual power series technique is introduced to obtain approximate solutions of such problems. The solution was obtained without reducing fractional differential equations to time fractional or space fractional differential equations. Some interesting results are presented to verify the efficiency and reliability of the developed algorithm.

Residual Power Series Technique for Simulating Fractional Bagley–Torvik Problems Emerging in Applied Physics

Numerical simulation of physical issues is often performed by nonlinear modeling, which typically involves solving a set of concurrent fractional differential equations through effective approximate methods. In this paper, an analytic-numeric simulation technique, called residual power series (RPS), is proposed in obtaining the numerical solution a class of fractional Bagley–Torvik problems (FBTP) arising in a Newtonian fluid. This approach optimizes the solutions by minimizing the residual error functions that can be directly applied to generate fractional PS with a rapidly convergent rate. The RPS description is presented in detail to approximate the solution of FBTPs by highlighting all the steps necessary to implement the algorithm in addressing some test problems. The results indicate that the RPS algorithm is reliable and suitable in solving a wide range of fractional differential equations applying in physics and engineering.

A reliable treatment of residual power series method for time-fractional Black–Scholes European option pricing equations

Approximate analytical solution of a fractional Black–Scholes pricing model is much relevant due to its practical importance to financial markets. In this article, a powerful approximate iterative mathematical scheme based on residual power series (RPS) algorithm is presented to achieve the numerical results of the nonlinear time fractional Black–Scholes (BS) equations based on European options. The Caputo-type fractional derivatives are considered in the present article. The residual power series method (RPSM) developed by Arqub (2013) is the novel technique for finding the analytical Taylor series solutions of systems of linear and nonlinear ODEs and PDEs. The residual power series technique supplies the approximate analytical solutions of the problem in truncated series form using residual error concept along with given initial conditions. The numerical procedures reveal that only a few iterations are sufficient for better approximations of the solutions, which clearly exhibits the reliability and effectiveness of this iterative scheme. This article shows that the adopted scheme is quite systematic as well as computationally attractive regarding solution procedure. In addition, effects of fractional order of time derivatives on the solutions for various particular cases are also depicted through graphs.

Symmetry analysis, exact solutions and numerical approximations for the space-time Carleman equation in nonlinear dynamical systems

This work presents an investigation and analysis for the space-time Carleman equation (STCE) in nonlinear dynamical systems. We compute the point symmetries, similarity variable, similarity transformation for STCE with Riemann-Liouville (RL) derivative and reduce STCE to ordinary differential equation (ODE) of fractional order. The exact solutions with conformable derivative are obtained via the generalized Bernoulli (GB) sub-ODE method. The well known residual power series technique (RPST) is used to compute the corresponding approximate solutions for the obtained exact solution. We then verify the convergence analysis and error estimate of RPST. Numerical simulations of the results are shown with the aid of graphical illustrations and tables. We prove that the RPST is very efficient for investigating the numerical approximations for a system of fractional differential equations.

Numerical Solutions of Linear Time-fractional Klein-Gordon Equation by Using Power Series Approach

In this paper, we provide approximate solution to linear time-fractional Kline-Gordon equations (FKGEs) with initial conditions by using the residual power series (RPS) method. The proposed technique relies on generalized Taylor formula under Caputo sense aiming at extracting a supportive analytical solution in convergent series form. Graphical results show the geometric behaviors to the approximate solutions at different values of fraction order 𝛾. The numerical analysis detects that the RPS technique is an efficient, simple and powerful tool to determine the solutions of the time-fractional KGE.

A residual power series technique for solving Boussinesq–Burgers equations

AbstractIn this paper, a residual power series method (RPSM) is combining Taylor’s formula series with residual error function, and is investigated to find a novel analytical solution of the coupled strong system nonlinear Boussinesq-Burgers equations according to the time. Analytical solution was purposed to find approximate solutions by RPSM and compared with the exact solutions and approximate solutions obtained by the homotopy perturbation method and optimal homotopy asymptotic method at different time and concluded that the present results are more accurate and efficient than analytical methods studied. Then, analytical simulations of the results are studied graphically through representations for action of time and accuracy of method.

A Residual Power Series Technique for Solving Systems of Initial Value Problems

In this article, a residual power series technique for the po wer series solution of systems of initial value problems is introduced. The new approach provides the solution in the form of a rapidly convergent series with easily computable components using symbolic computation software. The proposed technique obtains Taylor expansion of the solution of a system and reproduces the exact solution when the solution is polynomial. Numerical examples are included to demonstrate the efficiency, accu racy, and applicability of the presented technique. The results reve al that the technique is very effective, straightforward, a nd simple.