Power Series Method Research Articles

Numerical Analysis of Nonlinear Coupled Schrödinger–KdV System with Fractional Derivative

In this paper, we propose two efficient methods for solving the fractional-order Schrödinger–KdV system. The first method is the Laplace residual power series method (LRPSM), which involves expressing the solution as a power series and using residual correction to improve the accuracy of the solution. The second method is a new iterative method (NIM) that simplifies the problem and obtains a recursive formula for the solution. Both methods are applied to the Schrödinger–KdV system with fractional derivatives, which arises in many physical applications. Numerical experiments are performed to compare the accuracy and efficiency of the two methods. The results show that both methods can produce highly accurate solutions for the fractional Schrödinger–KdV system. However, the new iterative method is more efficient in terms of computational time and memory usage. Overall, our study demonstrates the effectiveness of the residual power series method and the new iterative method in solving fractional-order Schrödinger–KdV systems and provides a valuable tool for researchers and practitioners in applied mathematics and physics.

Group Analysis Explicit Power Series Solutions and Conservation Laws of the Time-Fractional Generalized Foam Drainage Equation

In this study, the classical Lie symmetry method is successfully applied to investigate the symmetries of the time-fractional generalized foam drainage equation with the Riemann–Liouville derivative. With the help of the obtained Lie point symmetries, the equation is reduced to nonlinear fractional ordinary differential equations (NLFODEs) which contain the Erdélyi–Kober fractional differential operator. The equation is also studied by applying the power series method, which enables us to obtain extra solutions. The obtained power series solution is further examined for convergence. Conservation laws for this equation are obtained with the aid of the new conservation theorem and the fractional generalization of the Noether operators.

Current density profiles at thin layer electrochemical cells under laminar and turbulent regimes

The transition of laminar to turbulent regimes on smooth electrocatalysts in thin-film electrochemical cells is a subject of study with the aim of predicting current density profiles. Analytical solutions are obtained for velocity profiles under steady state conditions for both hydrodynamic conditions applying a proper velocity contour condition on the catalyst surface. This state is necessary to describe electrochemical cell performances since in these cases, electrode reactions occur at the surface and not in the bulk of the electrolyte. The obtained equations for linear momentum under asymptotic conditions reduce to the solutions obtained by classical Power Series methods years ago. Concentration and current density profiles are also solved accordingly and compared with experimental data of fast redox and dissolved gas reactions on smooth noble metal electrocatalysts.

On the Solitary Waves and Nonlinear Oscillations to the Fractional Schrödinger–KdV Equation in the Framework of the Caputo Operator

The fractional Schrödinger–Korteweg-de Vries (S-KdV) equation is an important mathematical model that incorporates the nonlinear dynamics of the KdV equation with the quantum mechanical effects described by the Schrödinger equation. Motivated by the several applications of the mentioned evolution equation, in this investigation, the Laplace residual power series method (LRPSM) is employed to analyze the fractional S-KdV equation in the framework of the Caputo operator. By incorporating both the Caputo operator and fractional derivatives into the mentioned evolution equation, we can account for memory effects and non-local behavior. The LRPSM is a powerful analytical technique for the solution of fractional differential equations and therefore it is adapted in our current study. In this study, we prove that the combination of the residual power series expansion with the Laplace transform yields precise and efficient solutions. Moreover, we investigate the behavior and properties of the (un)symmetric solutions to the fractional S-KdV equation using extensive numerical simulations and by considering various fractional orders and initial fractional values. The results contribute to the greater comprehension of the interplay between quantum mechanics and nonlinear dynamics in fractional systems and shed light on wave phenomena and symmetry soliton solutions in such equations. In addition, the proposed method successfully solves fractional differential equations with the Caputo operator, providing a valuable computational instrument for the analysis of complex physical systems. Moreover, the obtained results can describe many of the mysteries associated with the mechanism of nonlinear wave propagation in plasma physics.

Exact solutions and invariant subspaces to the nonlinear dissipative–dispersive equation

In this paper, we performed Lie symmetry analysis and applied [Formula: see text] expansion method on the nonlinear dissipative–dispersive equation. The purpose of this research is to find the vector fields and transform the nonlinear dissipative–dispersive equation into simpler forms. The Maple software was used to obtain the vector field and similarity reductions for nonlinear dissipative–dispersive equations. In addition, we obtained exact solutions based on the [Formula: see text] expansion method and power series method, including the hyperbolic functions, the trigonometric functions and the rational functions. The method we used is direct, concise, elementary and effective, and can be used for many other nonlinear evolution equations. Furthermore, the invariant subspaces of the nonlinear dissipative–dispersive equation were identified using the refined invariant subspaces method. The invariant subspaces of solutions to linear ordinary differential equations were used to prove that nonlinear dissipative–dispersive equation admits subspaces. The exact solutions were obtained by using generalized separated variables.

Solution of conformable Laguerre and associated Laguerre equations using Laplace transform

In this paper, the conformable Laguerre and associated Laguerre differential equations are solved using the Laplace transform. The solution is found to be in exact agreement with that obtained using the power series method. In addition, some properties and some recursion relations of the Laguerre and associated Laguerre functions are discussed and proved. Then, the conformable Rodriguez’s formula and generating function are proposed.

Adaptation of Caputo residual power series scheme in solving nonlinear time fractional Schrödinger equations

This study regards the time-fractional nonlinear Schrödinger equations (TFNLSEs). The TFNLSE is used to depict nonlocal quantum phenomena in quantum physics and investigate the quantum behaviors of either long-range relations or time-dependent processes with many scales. Basic definitions of fractional derivatives are described in the Caputo fractional derivative (CaFD) sense. We utilize the residual power series method (RPSM) to achieve the approximate solutions of the TFNLSE. The RPSM is a numeric analytic technique for solving various ordinary, partial, and fuzzy differential and integrodifferential equations of fractional order. It is a practical optimization approach, supplying solutions in the closed form of known functions. The primary benefit of this technique is the clarity in calculating the coefficients of terms of the series solution by employing only the differential operators and not the other well-known analytic approaches that require integration operators, which are challenging in the fractional case. Our main aim in this paper is to find approximate solutions using RPSM and compare these approximate solutions with exact solutions. We discuss three applications, showing the validity, precision, and efficiency of the RPSM. The outcomes are presented in graphs and tables, which display the method’s effectiveness, quality, and strength. The RPSM is entirely compatible with the complexity of this problem, and the accepted outcomes are highly favorable. The offered technique can be easily used in the spaces of higher-dimension solutions without limiting the nature of the equation or the category style. Also, this method can be implemented as a novel alternative technique to deal with many nonlinear issues occurring in quantum physics.

A fractional-order model for computer viruses and some solution associated with residual power series method

Awareness of virus spreading is an important issue for building various defence strategies and protecting personal computers, smart devices, network devices, etc. In this research work, we develop epidemiological models to address this problem and introduce certain modified epidemiological fractional SAIR model, where we consider the fractional derivative in Caputo sense. We utilize the residual power series method to construct approximate solutions for the governing system. To show the efficiency and suitability of the proposed technique, we introduce a comparative between the obtained solutions and those solutions that are constructed using the fourth-order Runge–Kutta method. We derive some numerical results by considering specific values for the parameters in the governing model, and then we depict some of these results into two and three dimensions.

An effective analytical method for fractional Brusselator reaction–diffusion system

In recent years, reaction–diffusion models have attracted researchers for their wide applications. In this article, we consider Brusselator reaction–diffusion system (BRDS), which is known for its cross diffusion and pattern formations in biology and chemistry. We derive an analytical solution of the fractional Brusselator reaction–diffusion system (FBRDS) with the help of the initial condition by a novel method, residual power series method (RPSM). The system solution has been analyzed by graph.

An Exact In-Plane Equilibrium Equation for Transversely Loaded Large Deflection Membranes and Its Application to the Föppl-Hencky Membrane Problem

In the existing literature, there are only two in-plane equilibrium equations for membrane problems; one does not take into account the contribution of deflection to in-plane equilibrium at all, and the other only partly takes it into account. In this paper, a new and exact in-plane equilibrium equation is established by fully taking into account the contribution of deflection to in-plane equilibrium, and it is used for the analytical solution to the well-known Föppl-Hencky membrane problem. The power series solutions of the problem are given, but in the form of the Taylor series, so as to overcome the difficulty in convergence. The superiority of using Taylor series expansion over using Maclaurin series expansion is numerically demonstrated. Under the same conditions, the newly established in-plane equilibrium equation is compared numerically with the existing two in-plane equilibrium equations, showing that the new in-plane equilibrium equation has obvious superiority over the existing two. A new finding is obtained from this study, namely, that the power series method of using Taylor series expansion is essentially different from that of using Maclaurin series expansion; therefore, the recurrence formulas for power series coefficients of using Maclaurin series expansion cannot be derived directly from that of using Taylor series expansion.

Stress solution for an arbitrarily shaped tunnel with a neighboring circular cavity

This paper gives an analytical stress solution for the plane strain problem that the deep rock mass contains an arbitrarily shaped tunnel and a circular cavity subjected to in situ stresses. Based on the discovery that the mapping function for an arbitrarily shaped hole can also manage a tunnel and a circular hole mapping into two circles, the plate bounded by the two holes is conformally mapped onto an annulus. The complex variable method is used to establish the stress boundary condition, and then the two analytical functions are solved by the power series method. Comparison with ANSYS solution is made to validate the correctness of deductions and solving process. To consider practical underground engineering, the cases of different lateral stresses, distances of the two holes, sizes, and positions of the cavity are discussed.

Exact and Approximate Solutions for Linear and Nonlinear Partial Differential Equations via Laplace Residual Power Series Method

The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize the Laplace residual power series method to generate analytic solutions to various kinds of partial differential equations. Then, by resorting to the above-mentioned technique, we derive certain solutions to different types of linear and nonlinear partial differential equations, including wave equations, nonhomogeneous space telegraph equations, water wave partial differential equations, Klein–Gordon partial differential equations, Fisher equations, and a few others. Moreover, we numerically examine several results by investing some graphs and tables and comparing our results with the exact solutions of some nominated differential equations to display the new approach’s reliability, capability, and efficiency.

Statistical Convergence of Spliced Sequences in Terms of Power Series on Topological Spaces

In the present paper, $P-$distributional convergence which is defined by power series method has been introduced. We give equivalent expressions for $P-$distributional convergence of spliced sequences. Moreover, convergence of a bounded $\infty$-spliced sequence via power series method is represented in terms of Bochner integral in Banach spaces.

Approximate Solution of Schrodinger Equation to Diatomic Molecule for Harmonic Oscillator

This study has described the approximate solution of Schrodinger equation to diatomic molecule for harmonic oscillator. The solution procedure is developed by the Power series method and Newton’s second law. It consider an approximate solution of harmonic oscillator using Schrödinger equation in one dimension only because other analytical approaches are limited to the widely known method and consider two to five dimensions with various iteration method to obtain their results but here the solutions to be obtained and their efficiency will help other research to comprehend how the solution of this harmonic oscillator has been done over the years and also to use the most efficient approximate solution.

Approximation via statistical relative uniform convergence of sequences of functions at a point with respect to power series method

Approximation via statistical relative uniform convergence of sequences of functions at a point with respect to power series method

A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

AbstractArtificial neural networks have great potential for learning and stability in the face of tiny input data changes. As a result, artificial intelligence techniques and modeling tools have a growing variety of applications. To estimate a solution for fractal‐fractional differential equations (FFDEs) of high‐order linear (HOL) with variable coefficients, an iterative methodology based on a mix of a power series method and a neural network approach was applied in this study. In the algorithm's equation, an appropriate truncated series of the solution functions was replaced. To tackle the issue, this study uses a series expansion of an unidentified function, where this function is approximated using a neural architecture. Some examples were presented to illustrate the efficiency and usefulness of this technique to prove the concept's applicability. The proposed methodology was found to be very accurate when compared to other available traditional procedures. To determine the approximate solution to FFDEs‐HOL, the suggested technique is simple, highly efficient, and resilient.

Solution of time fractional Fitzhugh–Nagumo equation using semi analytical techniques

In this work, we apply three different techniques to solve the Fitzhugh–Nagumo equation that is an important equation used to describe the propagation of electrical signals in excitable media, such as nerve fibers. Residual power series method (RPSM), homotopy perturbation method (HPM), and a modified fractional Taylor expansion, are applied to this nonlinear equation to obtain approximate solutions. By comparing the exact solution with the approximate solutions obtained from the methods suggested we demonstrate that these methods are efficient tools to solve nonlinear fractional partial differential equations (NFPDE) this is due to the high accuracy obtained. To support the current solution investigation, various graphs in 2D and 3D are shown.

On statistical limit points with respect to power series methods and modulus functions

In this study, we define a new type of statistical limit point using the notions of statistical convergence with respect to the $J_p$ power series method and then we present some examples to show the relations between these points and ordinary limit points. After that we also study statistical limit points of a sequence with the help of a modulus function in the sense of the $J_p$ power series method. Namely, we define $f-J_p$-statistical limit and cluster points of the real sequences and compare the set of these limit points with the set of ordinary points.

Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations

In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m∈Z+, where the fractional derivative of order α=qγ, for which γ∈Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm’s accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of fractional problem. Under Caputo meaning of the symmetry order, the obtained results are illustrated numerically and graphically. Geometrically, the behavior of the obtained solutions declares that the changing of the fractional derivative parameter values in their domain alters the style of these solutions in a symmetric meaning, as well as indicates harmony and symmetry, which leads them to fully coincide at the value of the ordinary derivative. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for integral and integro-differential equations of fractional order.

Similarity Reductions, Power Series Solutions, and Conservation Laws of the Time-Fractional Mikhailov–Novikov–Wang System

The current study presents a comprehensive Lie symmetry analysis for the time-fractional Mikhailov–Novikov–Wang (MNW) system with the Riemann–Liouville fractional derivative. The corresponding simplified equations with the Erdélyi–Kober fractional derivative are constructed by group invariant solutions. Furthermore, we obtain explicit solutions with the help of the power series method and show the dynamical behavior via evolutional figures. Finally, by means of Ibragimov’s new conservation theorem, the conservation laws are derived for the system.