A. H. Bhrawy

• PHD
• Professor (Associate) at Beni-Suef University

This article adapts an operational matrix formulation of the collocation method for the one- and two-dimensional nonlinear fractional sub-diffusion equations (FSDEs). In the proposed collocation approach, the double and triple shifted Jacobi polynomials are used as base functions for approximate solutions of the one- and two-dimensional cases. The...

A shifted Legendre collocation method in two consecutive steps is developed and analyzed to numerically solve one-and two-dimensional time fractional Schrödinger equations (TFSEs) subject to initial-boundary and non-local conditions. The first step depends mainly on shifted Legendre Gauss–Lobatto collocation (SL-GL-C) method for spatial discretizat...

The cable equation plays a central role in many areas of electrophysiology and in modeling neuronal dynamics. This paper reports an accurate spectral collocation method for solving one- and twodimensional variable-order fractional nonlinear cable equations. The proposed method is based on shifted Jacobi collocation procedure in conjunction with the...

As a natural generalization of the fractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena. In this paper, we develop an exponentially accurate Jacobi–Gauss–Lobatto collocation (J–GL-C) method to solve the variable-order fractional Schrödinger equations in one dime...

This paper presents a new pseudospectral technique to solve three dimensional integral equations (3D-IEs). The shifted Legendre Gauss-Lobatto collocation method is investigated to approximate the 3D-IEs. The main idea in the novel algorithm is to reduce the 3D-IEs to systems of algebraic equations. The applicability of the present method is examine...

In this paper, we propose a new accurate and robust numerical technique to approximate the solutions of fractional variational problems (FVPs) depending on indefinite integrals with a type of fixed Riemann–Liouville fractional integral. The proposed technique is based on the shifted Chebyshev polynomials as basis functions for the fractional integr...

The operational matrices of left Caputo fractional derivative, right Caputo fractional derivative, and Riemann–Liouville fractional integral, for shifted Chebyshev polynomials, are presented and derived. We propose an accurate and efficient spectral algorithm for the numerical solution of the two-sided space–time Caputo fractional-order telegraph e...

This paper reports a new numerical approach for numerically solving types of fractional variational problems. In our approach, we use the fractional integrals operational matrix, described in the sense of Riemann–Liouville, with the help of the Lagrange multiplier technique for converting the fractional variational problem into an easier problem th...

A distributed-order time- and Riesz space-fractional Schrödinger equation (DOT–RSFSE) is considered. Distributed-order derivatives indicate fractional derivatives that are integrated over the order of the differentiation within a given range. That is to say, the order of the time derivative ranges from zero to one. The space-fractional derivative i...

This paper addresses the solution of one- and two-dimensional Volterra integral equations (VIEs) by means of the spectral collocation method. The novel technique takes advantage of the properties of shifted Jacobi polynomials and is applied for solving multi-dimensional VIEs. Several numerical examples demonstrate the efficiency of the method and a...

In this paper, we applied a direct method for a solution of isoperimetric fractional variational problems. We use shifted Legendre orthonormal polynomials as basis function of operational matrices of fractional differentiation and fractional integration in combination with the Lagrange multipliers technique for converting such isoperimetric fractio...

In this manuscript we report a new operational technique for approximating the numerical solution of fractional optimal control (FOC) problems. The operational matrix of the Caputo fractional derivative of the orthonormal Chebyshev polynomial and the Legendre-Gauss quadrature formula are used, and then the Lagrange multiplier scheme is employed for...

This paper adapts a new numerical technique for solving two-dimensional fractional integral equations with weakly singular. Using the spectral collocation method, fractional operators of Legendre and Chebyshev polynomials, and Gauss-quadrature formula, we achieve a reduction of such problems into those of a system of algebraic equations. For testin...

Current discretizations of variable-order fractional (V-OF) differential equations lead to numerical solutions of low order of accuracy. This paper explores a high order numerical scheme for multi-dimensional V-OF Schrödinger equations. We derive new operational matrices for the V-OF derivatives of Caputo and Riemann–Liouville type of the shifted J...

In this paper, a new space-time spectral algorithm is constructed to solve the generalized Hirota-Satsuma coupled Korteweg-de Vries (GHS-C-KdV) system of time-fractional order. The present algorithm consists of applying the collocation-spectral method in conjunction with the operational matrix of fractional derivative for the double Jacobi polynomi...

The mixed Volterra-Fredholm integral equations (VFIEs) arise in various physical and biological models. The main purpose of this article is to propose and analyze efficient Bernoulli collocation techniques for numerically solving classes of two-dimensional linear and nonlinear mixed VFIEs. The novel aspect of the technique is that it reduces the pr...

The space-time fractional diffusion-wave equation (FDWE) is a generalization of classical diffusion and wave equations which is used in modeling practical phenomena of diffusion and wave in fluid flow, oil strata and others. This paper reports an accurate spectral tau method for solving the two-sided space and time Caputo FDWE with various types of...

The operational matrices of left Caputo fractional derivative, right Caputo fractional derivative, and Riemann–Liouville fractional integral, for shifted Chebyshev polynomials, are presented and derived. We propose an accurate and efficient spectral algorithm for the numerical solution of the two-sided space–time Caputo fractional-order telegraph e...

The variable order wave equation plays a major role in acoustics, electromagnetics, and fluid dynamics. In this paper, we consider the space–time variable order fractional wave equation with variable coefficients. We propose an effective numerical method for solving the aforementioned problem in a bounded domain. The shifted Jacobi polynomials are...

Because of the non-local properties of fractional operators, obtaining the analytical solutions of partial differential equations with fractional variable order is more challenging. Efficiently solving these equations naturally becomes an urgent topic. This paper reports an efficient numerical solution of the Rayleigh-Stokes (R-S) problem with vari...

A spectral shifted Legendre Gauss–Lobatto collocation method is developed and analyzed to solve numerically one-dimensional two-sided space fractional Boussinesq (SFB) equation with non-classical boundary conditions. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials (Formula presented.) is assume...

While several high-order methods have been extensively developed for fixed-order fractional differential equations (FDEs), there are no such methods for variable-order FDEs. In this paper, we propose an accurate and robust approach to approximate the solution of functional Dirichlet boundary value problem with a type of variable-order Caputo fracti...

This paper derives a new operational matrix of the variable-order (VO) time fractional partial derivative involved in anomalous diffusion for shifted Chebyshev polynomials. We then develop an accurate numerical algorithm to solve the 1+1 and 2+1 VO and constant-order fractional diffusion equation with Dirichlet conditions. The contraction of the pr...

This paper applies the well known G′/G-expansion scheme to obtain dark and singular optical soliton solutions in optical fiber with parabolic law nonlinearity in presence of external potential and Raman scattering. Time-dependent coefficients are taken into consideration. There are constraint conditions, also known as integrability criteria, that n...

This paper studies dispersive solitons in optical nanofibers that are modeled by Schrödinger-Hirota equation. The tanh-coth integration algorithm obtains soliton solutions to the model that are studied with Kerr law and power law nonlinearity. There are constraint conditions that fall out for these solitons to exist.

In the present paper, we construct the numerical solution for time fractional (1 + 1)- and (1 + 2)-dimensional Schrödinger equations (TFSEs) subject to initial boundary. The solution is expanded in a series of shifted Jacobi polynomials in time and space. A collocation method in two steps is developed and applied. First step depends mainly on appli...

This paper utilizes trial solution algorithm to secure optical soliton solutions to the nonlinear Schrödinger’s equation. Bright, dark and singular soliton solutions are obtained to the model that is considered with four forms of nonlinearity. They are Kerr, power, parabolic and dual-power laws. There are constraint conditions that guarantee the ex...

This paper reports a new Legendre–Gauss–Lobatto collocation (SL-GL-C) method to solve numerically two partial parabolic inverse problems subject to initial-boundary conditions. The problem is reformulated by eliminating the unknown functions using some special assumptions based on Legendre–Gauss—Lobatto quadrature rule. The SL-GL-C is utilized to s...

This paper reports a new spectral collocation technique for solving second kind Fredholm integral equations (FIEs). We develop a collocation scheme to approximate FIEs by means of the shifted Legendre–Gauss–Lobatto collocation (SL–GL-C) method. Moreover, we adapt the SL–GL-C algorithm to solve one dimensional second kind FIEs and system of FIEs. Tw...

This paper studies the application of tanh method to address a few coupled nonlinear evolution equations that are in complex domain. There are soliton solutions as well as triangular solutions that are revealed with this integration scheme. The equations studied in this paper are applicable to various branches of applied and theoretical physics.

This paper reports new modified Jacobi polynomials (MJPs). We derive the basis transformation between MJPs and Bernstein polynomials and vice versa. This transformation is merging the perfect Least-square performance of the new polynomials together with the geometrical insight of Bernstein polynomials. The MJPs with indexes corresponding to the num...

A shifted Jacobi collocation method in two stages is constructed and used to numerically solve nonlinear Schrödinger equations (NLSEs) with a Kerr law nonlinearity, subject to initial-boundary conditions. An expansion in a series of spatial shifted Jacobi polynomials with temporal coefficients for the approximate solution is considered. The first s...

The vector-coupled nonlinear Schrödinger equation, which can be applied to describe the propagation of Thirring optical solitons in birefringent fibers with Kerr law nonlinearity, detuning, intermodal dispersion and spatiotemporal dispersion, has been studied analytically. By means of the complex envelope function ansatz, exact Thirring bright-dark...

This paper reports a new spectral collocation technique for solving time-space modified anomalous subdiffusion equation with a nonlinear source term subject to Dirichlet and Neumann boundary conditions. This model equation governs the evolution for the probability density function that describes anomalously diffusing particles. Anomalous diffusion...

The study of shallow water waves, in (1+1)-dimensions, governed by the Korteweg-de Vries (KdV) equation, with logarithmic-law nonlinearity, is conducted in this paper. Exact Gaussian solitary wave solutions are obtained using three integration schemes. The conservation laws are listed. The adiabatic parameter dynamics is also given in presence of p...

In this work, we study the dynamics of optical solitons in a synthetic nonlocal nonlinear media. The nonlinear dynamical model which describes the propagation of optical solitons in the weakly nonlocal nonlinear media with parabolic law nonlinearity is investigated analytically. The tool of integration that is the Riccati equation mapping approach...

This paper reports conservation laws for coupled wave equations that are studied in several contexts. These include two-layered shallow water waves, long-short wave interactions, longitudinal and transverse wave interactions and others. The conserved densities are secured with the aid of Lie symmetry analysis, while the conserved quantities are obt...

This paper reports a new fully collocation algorithm for the numerical solution of hyperbolic partial differential equations of second order in a semi-infinite domain, using Jacobi rational Gauss-Radau collocation method. The widely applicable, efficiency, and high accuracy are the key advantages of the collocation method. The series expansion in J...

In this paper we present the operational matrices of the left Caputo fractional derivative, right Caputo fractional derivative and Riemann–Liouville fractional integral for shifted Legendre polynomials. We develop an accurate numerical algorithm to solve the two-sided space–time fractional advection–dispersion equation (FADE) based on a spectral sh...

This paper reports a new numerical method that enables easy and convenient discretization of a two-dimensional sub-diffusion equation with fractional derivatives of any order. The suggested method is based on Jacobi tau spectral procedure together with the Jacobi operational matrix for fractional derivatives, described in the Caputo sense. Such app...

This paper reports a novel numerical technique for solving the time variable fractional order mobile-immobile advection-dispersion (TVFO-MIAD) model with the Coimbra variable time fractional derivative, which is preferable for modeling dynamical systems. The main advantage of the proposed method is that two different collocation schemes are investi...

In this paper, the Chebyshev Gauss-Lobatto pseudospectral scheme is investigated in spatial directions for solving one-dimensional, coupled, and two-dimensional parabolic partial differential equations with time delays. For the one-dimensional problem, the spatial integration is discretized by the Chebyshev pseudospectral scheme with Gauss-Lobatto...

This paper reports new orthogonal functions on the half line based on the definition of the classical Jacobi polynomials. We derive an operational matrix representation for the differentiation of exponential Jacobi functions which is used to create a new exponential Jacobi pseudospectral method based on the operational matrix of exponential Jacobi...

This paper obtains soliton solutions to nonlinear Schrödinger's equation with quadratic nonlinearity. There are five integration schemes that are applied to retrieve these soliton solutions. These are Q-function method, G' /G-expansion scheme, Riccati equation approach and finally the mapping method along with the modified mapping method. The const...

In this paper, new operational matrices for shifted Legendre orthonormal polynomial are derived. This polynomial is used as a basis function for developing a new numerical technique for the delay fractional optimal control problem. The fractional integral is described in the Riemann–Liouville sense, while the fractional derivative is described in t...

We propose a new efficient spectral collocation method for solving a time fractional sub-diffusion equation on a semi-infinite domain. The shifted Chebyshev-Gauss-Radau interpolation method is adapted for time discretization along with the Laguerre-Gauss-Radau collocation scheme that is used for space discretization on a semi-infinite domain. The m...

In this paper, we derive a Gauss-Lobatto collocation method to solve numerically parabolic equations subject to initial-integro conditions. The spatial approximation is based on a Gauss-Lobatto collocation method. Here, we use the Legendre polynomials as the space basis functions. The Legendre Gauss-Lobatto quadrature rule is investigated for treat...

In this article, we develop a numerical approximation for first-order hyperbolic equations on semi-infinite domains by using a spectral collocation scheme. First, we propose the generalized Laguerre-Gauss-Radau collocation scheme for both spatial and temporal discretizations. This in turn reduces the problem to the obtaining of a system of algebrai...

In this paper, we derive and analyze an efficient spectral collocation algorithm to solve numerically some wave equations subject to initial-boundary nonlocal conservation conditions in one and two space dimensions. The Legendre pseudospectral approximation is investigated for spatial approximation of the wave equations. The Legendre–Gauss– Lobatto...

This paper extends the application of the spectral Jacobi-Gauss-Lobatto collocation (J-GL-C) method based on Gauss-Lobatto nodes to obtain semi-analytical solutions of nonlinear time-dependent reaction-diffusion equations (RDEs) subject to Dirichlet boundary conditions. This approach has the advantage of allowing us to obtain the solution in terms...

This paper reports two successive spectral collocation methods, that enable easy and highly accuracy discretization, for 1 + 1 and 2 + 1 fractional percolation equations (FPEs). The first step depends mainly on the shifted Legendre Gauss–Lobatto collocation method for spatial discretization. An expansion in a series of shifted Legendre polynomials...

In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absol...

This paper presents a shifted fractional-order Jacobi orthogonal function (SFJF) based on the definition of the classical Jacobi polynomial. A new fractional integral operational matrix of the SFJF is presented and derived. We propose the spectral Tau method, in conjunction with the operational matrices of the Riemann–Liouville fractional integral...

In this article, we construct a new numerical approach for solving the time-fractional Fokker–Planck equation. The shifted Jacobi polynomials are used as basis functions, and the fractional derivative is described in the sense of Caputo. The proposed approach is a combination of shifted Jacobi Gauss–Lobatto scheme for the spatial discretization and...

The numerical solution of a fractional optimal control problem having a quadratic performance index is proposed and analyzed. The performance index of the fractional optimal control problem is considered as a function of both the state and the control variables. The dynamic constraint is expressed as a fractional differential equation that includes...

Recently, operational matrices were adapted for solving several kinds of fractional differential equations (FDEs). The use of numerical techniques in conjunction with operational matrices of some orthogonal polynomials, for the solution of FDEs on finite and infinite intervals, produced highly accurate solutions for such equations. This article dis...

In this paper, a high accurate numerical approach is investigated for solving the time-fractional linear and nonlinear Korteweg-de Vries (KdV) equations. These equations are the most appropriate and desirable definition for physical modeling. The spectral collocation method and the operational matrix of fractional derivatives are used together with...

This paper studies optical solitons with parabolic law nonlinearity in presence of nonlinear dispersion. There are three integration tools that are adopted to retrieve soliton solutions. They are the Jacobi's elliptic function method, Riccati's equation approach, and the ansatz scheme. Bright, dark as well as singular solitons are obtained along wi...

This paper obtains bright 1-soliton solutions in optical metamaterials by the aid of traveling wave hypothesis. There are three types of nonlinear media that are considered. They are Kerr law, parabolic law and log law nonlinearity. There are several constraint relations that are obtained for soliton solutions to exist. © 2015, National Institute o...

In this paper, we are concerned with the fractional sub-diffusion equation with a non-linear source term. The Legendre spectral collocation method is introduced together with the operational matrix of fractional derivatives (described in the Caputo sense) to solve the fractional sub-diffusion equation with a non-linear source term. The main charact...

This paper obtains bright, dark and singular soliton solutions to dense wavelength division multiplexed (DWDM) system, with spatio-temporal dispersion. There are two types of nonlinear media that are considered and they are Kerr law and parabolic law. Four integration algorithms are applied to retrieve these solitons. They are G′/G-expansion scheme...

In this study, we propose shifted fractional-order Jacobi orthogonal functions (SFJFs) based on the definition of the classical Jacobi polynomials. We derive a new formula that explicitly expresses any Caputo fractional-order derivatives of SFJFs in terms of the SFJFs themselves. We also propose a shifted fractional-order Jacobi tau technique based...

The dynamics of optical solitons in birefringent fibers are addressed in this paper. The models are studied with both Kerr and parabolic laws of nonlinearity without four-wave mixing terms. The integration tool is the Lie symmetry approach. Similarity reductions to the derived ordinary differential equations are investigated by Lie classical method...

The shifted Legendre orthogonal polynomials are used for the numerical solution of a new formulation for the multi-dimensional fractional optimal control problem (M-DFOCP) with a quadratic performance index. The fractional derivatives are described in the Caputo sense. The Lagrange multiplier method for the constrained extremum and the operational...

In this article, we introduce a numerical technique for solving a general form of the fractional optimal control problem. Fractional derivatives are described in the Caputo sense. Using the properties of the shifted Jacobi orthonormal polynomials together with the operational matrix of fractional integrals (described in the Riemann-Liouville sense)...

This work studies nonlinear dynamics of optical solitons in a cascaded system with Kerr law nonlinearity and spatio-temporal dispersion. The mathematical model that describes the propagation of optical solitons through a cascaded system is given by the vector-coupled nonlinear Schrödinger equation. It is investigated analytically using three integr...

The nonlinear dynamics of optical solitons with spatio-temporal dispersion, nonlinear dispersion and inter-modal dispersion have been investigated. The well-posed nonlinear Schrodinger equation is solved analytically suing the complex envelope function ansatz. The combined solitons are obtained. Finally, the effects of nonlinear dispersion as well...

Soliton solutions in DWDM system is recovered in presence of four-wave mixing terms. The two laws of nonlinearity considered are Kerr law and parabolic law. Exact bright, dark and singular soliton solutions are retrieved by the aid of ansatz method. This integrability is achieved only with phase-matching condition for these components.

This paper obtains soliton solutions in cascaded system with Kerr law nonlinearity. There are three integration tools adopted in this paper. These are Q-function approach, Riccati equation method and GIG-expansion scheme. These lead to topological and singular optical soliton solutions to the model. Additionally, there are singular periodic solutio...

The time-fractional coupled Korteweg–de Vries (KdV) system is a generalization of the classical coupled KdV system and obtained by replacing the first order time derivatives by fractional derivatives of orders \(\nu _1\) and \(\nu _2\), \((0<\nu _1,\nu _2\le 1).\) In this paper, an accurate and robust numerical technique is proposed for solving the...

The screening of the Coulomb potential inside heavy neutral atoms is modeled by Thomas-Fermi equation. In this paper, we propose a new numerical scheme to obtain an approximate solution of this equation on a semi-infinite interval. The proposed method is based upon the Jacobi rational functions in conjunction with Gauss quadrature formula. The prob...

Chapter 11 is devoted to numerical solutions of fractional differential equations (FDEs) on a semi-infinite interval. This chapter presents a broad discussion of spectral techniques based on operational matrices of fractional derivatives and integration methods for solving several kinds of linear and nonlinear FDEs. We present the operational matri...

This paper studies complex-valued Klein-Gordon equation that arises in field theory. Three integration tools are availed of in order to obtain soliton and other solutions to the governing equation that is considered with cubic and power law nonlinearity. The three algorithms that are studied in this paper are G'/G-expansion scheme, Kudryashov's met...

This paper studies the long-wave short-wave interaction equation that produces soliton solutions as well as other exact solutions. Exact 1-soliton solutions are obtained for this equation, with time-dependent coefficients, by the aid of ansatz method. Subsequently, the simplest equation approach also gives soliton solutions as well as other solutio...

Burgers’ equation is a fundamental partial differential equation in fluid mechanics. This paper reports a new space-time spectral algorithm for obtaining an approximate solution for the space-time fractional Burgers’ equation (FBE) based on spectral shifted Legendre collocation (SLC) method in combination with the shifted Legendre operational matri...

In this article, we propose the shifted Legendre orthonormal polynomials for the numerical solution of the fractional optimal control problems that appear in several branches of physics and engineering. The Rayleigh-Ritz method for the necessary conditions of optimization and the operational matrix of fractional derivatives are used together with t...

This paper reports a new spectral collocation algorithm for solving time-space fractional partial differential equations with subdiffusion and superdiffusion. In this scheme we employ the shifted Legendre Gauss-Lobatto collocation scheme and the shifted Chebyshev Gauss-Radau collocation approximations for spatial and temporal discretizations, respe...

The integrability aspects of complex-valued Klein-Gordon equation is studied in this paper. Three integration algorithms applied. Several solutions are obtained using such integration machineries. Solutions range from plane waves to shock waves and solitons as well as singular periodic solutions.

This paper presents an approach for solving fractional differential equations by employing the exp-function method and (G'/G)-expansion method. These methods were applied in two examples to solve non-linear fractional differential equations. The fractional derivatives are described in the modified Riemann-Liouville sense. As a result, many exact an...

In this paper, the cascaded system is revisited with Kerr and power laws of nonlinearity. The spatio-temporal dispersion is included this time in order to make the model of study well-posed. Bright, dark and singular soliton solutions are obtained along with respective constraints. These integrability conditions must hold for the solitons to exist...

The dynamics of Thirring optical solitons, with spatio-temporal dispersion, is studied in this paper. Bright, dark, and singular optical solitons are obtained by the ansatz method. There are constraint conditions that naturally emerge from the mathematical analysis.

In this paper, we derive an efficient spectral collocation algorithm to solve numerically the nonlinear complex generalized Zakharov system (GZS) subject to initial-boundary conditions. The Jacobi pseudospectral approximation is investigated for spatial approximation of the GZS. It possesses the spectral accuracy in space. The Jacobi–Gauss–Lobatto...

In this paper, a new spectral collocation method is applied to solve Lane-Emden equations on a semi-infinite domain. The method allows us to overcome difficulty in both the nonlinearity and the singularity inherent in such problems. This Jacobi rational-Gauss method, based on Jacobi rational functions and Gauss quadrature integration, is implemente...

This paper formulates a new explicit expression for the generalized Jacobi polynomials (GJPs) in terms of Bernstein basis. We also establish and prove the basis transformation between the GJPs basis and Bernstein basis and vice versa. This transformation embeds the perfect least-square performance of the GJPs with the geometrical insight of the Ber...

The Jacobi spectral collocation method (JSCM) is constructed and used in combination with the operational matrix of fractional derivatives (described in the Caputo sense) for the numerical solution of the time-fractional Schrödinger equation (T-FSE) and the space-fractional Schrödinger equation (S-FSE). The main characteristic behind this approach...

The Jacobi spectral collocation method (JSCM) is constructed and used in combination with the operational matrix of fractional derivatives (described in the Caputo sense) for the numerical solution of the time-fractional Schrödinger equation (T-FSE) and the space-fractional Schrödinger equation (S-FSE). The main characteristic behind this approach...

In this article, we present a numerical approximation of the initial-boundary system of nonlinear hyperbolic equations based on spectral Jacobi-Gauss-Radau collocation (J-GR-C) method. A J-GR-C method in combination with the implicit Runge-Kutta scheme are employed to obtain a highly accurate approximation to the mentioned problem. J-GR-C method, b...