Showing papers by "Zakieh Avazzadeh published in 2020"
TL;DR: The transcendental Bernstein series is presented as a generalization of the classical Bernstein polynomials for solving the variable-order space–time fractional telegraph equation (V-STFTE) using an approximation method using optimization techniques and the TBS.
Abstract: This paper presents the transcendental Bernstein series (TBS) as a generalization of the classical Bernstein polynomials for solving the variable-order space–time fractional telegraph equation (V-STFTE). An approximation method using optimization techniques and the TBS is introduced. The solution of the problem under consideration is expanded in terms of TBS with unknown free coefficients and control parameters. The new corresponding operational matrices of variable-order fractional derivatives, in the Caputo type, are derived. The proposed approach reduces the V-STFTE to a system of algebraic equations and, subsequently, to find the free coefficients and control parameters using the Lagrange multipliers technique. The convergence analysis of the method is guranteed by means of a new theorem concerning the TBS. The experimental results confirm the high accuracy and computational efficiency of the TBS method.
39 citations
TL;DR: In this article, the authors deal with approximating the time fractional Tricomi-type model in the sense of the Caputo derivative, where the spatial discretization is obtained using the local radial basis function in a finite difference mode.
34 citations
TL;DR: This paper studies the second kind linear Volterra integral equations with a discontinuous kernel obtained from the load leveling and energy system problems with the homotopy perturbation method (HPM), and takes advantage of the discrete stochastic arithmetic (DSA) to find the optimal iteration, optimal error and optimal approximation of the HPM.
Abstract: This paper studies the second kind linear Volterra integral equations (IEs) with a discontinuous kernel obtained from the load leveling and energy system problems. For solving this problem, we propose the homotopy perturbation method (HPM). We then discuss the convergence theorem and the error analysis of the formulation to validate the accuracy of the obtained solutions. In this study, the Controle et Estimation Stochastique des Arrondis de Calculs method (CESTAC) and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library are used to control the rounding error estimation. We also take advantage of the discrete stochastic arithmetic (DSA) to find the optimal iteration, optimal error and optimal approximation of the HPM. The comparative graphs between exact and approximate solutions show the accuracy and efficiency of the method.
33 citations
TL;DR: A goodness of fit test for discrete-time almost cyclostationary models based on the spectral support estimation and the application of multiple testing is presented.
33 citations
26 citations
19 Sep 2020
TL;DR: In this article, the Rosenau-Kortewegde Vries regularized-long wave equation (RK-LWE) is approximated by a local meshless technique called radial basis function (RBF) and the finite-difference method.
Abstract: This paper investigates the solitary wave solutions of the generalized Rosenau–Korteweg-de Vries-regularized-long wave equation. This model is obtained by coupling the Rosenau–Korteweg-de Vries and Rosenau-regularized-long wave equations. The solution of the equation is approximated by a local meshless technique called radial basis function (RBF) and the finite-difference (FD) method. The association of the two techniques leads to a meshless algorithm that does not requires the linearization of the nonlinear terms. First, the partial differential equation is transformed into a system of ordinary differential equations (ODEs) using radial kernels. Then, the ODE system is solved by means of an ODE solver of higher-order. It is shown that the proposed method is stable. In order to illustrate the validity and the efficiency of the technique, five problems are tested and the results compared with those provided by other schemes.
22 citations
TL;DR: In this article, an operational matrix (OM) scheme based on the shifted Chebyshev cardinal functions (SCCFs) of the second kind for numerical solution of the variable-order time fractional nonlinear reaction diffusion equation was proposed.
Abstract: This paper is concerned with an operational matrix (OM) scheme based on the shifted Chebyshev cardinal functions (SCCFs) of the second kind for numerical solution of the variable-order time fractional nonlinear reaction–diffusion equation. The fractional derivative operator is defined in the sense of Atangana–Baleanu–Caputo. Through the way, a new OM of variable-order fractional derivative is derived for the mentioned cardinal functions. More precisely, the unknown solution is expanded by the SCCFs with undetermined coefficients. Then the expansion substituted in the equation and the generated OM is utilized to extract some algebraic equations. The precision of the established approach is examined through various types of test examples. Numerical simulations confirm the suggested approach is highly accurate to provide satisfactory results.
21 citations
TL;DR: In this paper, a new version of the strongly coupled nonlinear fractal-fractional Schrodinger equations is introduced by using the fractal fractional derivatives in the Riemann-Liouville sense with Mittag-Leffler kernel and an accurate operational matrix method based on shifted Chebyshev cardinal functions is established for solving this new class of problems.
Abstract: In this paper, a new version of the strongly coupled nonlinear fractal-fractional Schrodinger equations is introduced by using the fractal-fractional derivatives in the Riemann-Liouville sense with Mittag-Leffler kernel. An accurate operational matrix method based on the shifted Chebyshev cardinal functions is established for solving this new class of problems. Along the way, a new operational matrix of fractal-fractional derivative is derived for these basis functions. The main characteristic of the proposed method is that it transforms solving the original problem to an algebraic system of equations by exploiting the operational matrix techniques.
21 citations
TL;DR: The solution of the NVOFPDE is expanded following the GSCP and the corresponding operational matrices of VO fractional derivatives, in the Caputo type, are obtained and an optimization method converts the problem into a system of nonlinear algebraic equations.
21 citations
TL;DR: In this paper, an efficient numerical approach is formulated for solving a category of non-singular variable-order time fractional coupled Burgers' equations with the aid of the Hahn polynomials.
Abstract: In this study, an efficient numerical approach is formulated for solving a category of non-singular variable-order time fractional coupled Burgers’ equations with the aid of the Hahn polynomials. The fractional differential operators are considered in the Atangana–Baleanu–Caputo concept. The designed method converts the original system into an algebraic system which can be simply handled. In order to verify that the demonstrated algorithm is reliable and accurate, some numerical experiments have been processed. The obtained solutions manifest the effectiveness and accuracy of the presented method for solving this class of equations.
20 citations
TL;DR: In this paper, the nonlinear space-time fractal-fractional advection-diffusion-reaction equation is introduced and a highly accurate methodology is presented for its numerical solution.
Abstract: In this paper, the nonlinear space–time fractal–fractional advection–diffusion–reaction equation is introduced and a highly accurate methodology is presented for its numerical solution. In the time...
TL;DR: The main ideas of this approach are to expand the unknown functions in tems of the shifted second-kind CCFs and apply the collocation method such that it reduces the problem into a system of algebraic equations.
Abstract: In this study, a computational approach based on the shifted second-kind Chebyshev cardinal functions (CCFs) is proposed for obtaining an approximate solution of coupled variable-order time-fractional sine-Gordon equations where the variable-order fractional operators are defined in the Caputo sense. The main ideas of this approach are to expand the unknown functions in tems of the shifted second-kind CCFs and apply the collocation method such that it reduces the problem into a system of algebraic equations. To algorithmize the method, the operational matrix of variable-order fractional derivative for the shifted second-kind CCFs is derived. Meanwhile, an effective technique for simplification of nonlinear terms is offered which exploits the cardinal property of the shifted second-kind CCFs. Several numerical examples are examined to verify the practical efficiency of the proposed method. The method is privileged with the exponential rate of convergence and provides continuous solutions with respect to time and space. Moreover, it can be adapted for other types of variable-order fractional problems straightforwardly.
TL;DR: In this article, a semi-discrete method based on 2D Chelyshkov polynomials (CPs) was developed to provide an approximate solution of the fractal-fractional nonlinear Emden-Fowler equation.
Abstract: This paper develops an effective semi-discrete method based on the 2D Chelyshkov polynomials (CPs) to provide an approximate solution of the fractal–fractional nonlinear Emden–Fowler equation. In t...
TL;DR: In this article, a new version of the nonlinear space-time fractional KdV-Burgers-Kuramoto equation has been generated via the variable-order (VO) fractional derivatives defined in the Caputo type.
Abstract: In this paper, a new version of the nonlinear space-time fractional KdV–Burgers–Kuramoto equation has been generated via the variable-order (VO) fractional derivatives defined in the Caputo type. A numerical method has been developed based on the discrete Legendre polynomials (LPs) and the collocation scheme for solving this equation. First, the solution of the problem is expanded in terms of the shifted discrete LPs. Then, this expansion and its derivatives, including the classical partial derivatives and the VO fractional partial derivatives are replaced in the equation. Eventually, the operational matrices of the shifted discrete LPs, including the classical derivatives and the VO fractional derivatives (which are derived in this study), and the collocation method are employed to convert the approximated problem into an algebraic system of equations. Some numerical results are given to illustrate the accuracy of the method.
TL;DR: In this paper, a nonlinear variable-order (VO) time fractional 2D Ginzburg-Landau equation was introduced by replacing the conventional derivative with the Atangana-Baleanu-Caputo fractional derivative.
Abstract: This paper introduces the nonlinear variable-order (VO) time fractional 2D Ginzburg-Landau equation by replacing the conventional derivative with the Atangana–Baleanu–Caputo fractional derivative. An efficient moving least squares (MLS) meshfree approximation method is considered to devise an algorithm for solving this enhanced equation. To be precise, first, the finite difference scheme is used to evaluate the fractional differentiation. Then, a recurrence formula is derived by applying the θ-weighted technique. Next, the real and imaginary components of the solution are expanded in terms of meshless functions. Last, these expansions which include unknown coefficients are input in the original equation. Therefore, the equation is converted into a system of linear algebraic equations which is uncomplicated for solving by mathematical software. To verify the validity of the devised method and demonstrate its precision, several problems are put to the test.
TL;DR: This study advances variable-order (VO) fractional delay differential models in the pantograph type introduced in the Caputo sense by employing the Chebyshev cardinal functions including unknown coefficients to construct a system with algebraic equations.
Abstract: This study advances variable-order (VO) fractional delay differential models in the pantograph type introduced in the Caputo sense. A method utilizing the Chebyshev cardinal functions (CCFs) is formulated to find an accurate result. In the proposed scheme, we first expand the solution in terms of the CCFs including unknown coefficients. Then, by inserting this expansion into the problem, employing the VO fractional differentiation operational matrix (OM) and the delay OM, we construct a system with algebraic equations. The generated algebraic systems are mainly sparse due to the cardinality of these functions and it significantly reduces the cost of computation.
TL;DR: In this article, a wavelet method is developed to solve a system of nonlinear variable-order (V-O) fractional integral equations using the Chebyshev wavelets (CWs) and the Galerkin method.
Abstract: In this study, a wavelet method is developed to solve a system of nonlinear variable-order (V-O) fractional integral equations using the Chebyshev wavelets (CWs) and the Galerkin method. For this purpose, we derive a V-O fractional integration operational matrix (OM) for CWs and use it in our method. In the established scheme, we approximate the unknown functions by CWs with unknown coefficients and reduce the problem to an algebraic system. In this way, we simplify the computation of nonlinear terms by obtaining some new results for CWs. Finally, we demonstrate the applicability of the presented algorithm by solving a few numerical examples.
TL;DR: This work develops an optimization method based on a new class of basis function, namely the generalized Bernoulli polynomials (GBP), to solve a class of nonlinear 2-dim fractional optimal control problems.
Abstract: This work develops an optimization method based on a new class of basis function, namely the generalized Bernoulli polynomials (GBP), to solve a class of nonlinear 2-dim fractional optimal control problems. The problem is generated by nonlinear fractional dynamical systems with fractional derivative in the Caputo type and the Goursat–Darboux conditions. First, we use the GBP to approximate the state and control variables with unknown coefficients and parameters. Afterwards, we substitute the obtained values for the variables and parameters in the objective function, nonlinear fractional dynamical system and Goursat–Darboux conditions. The 2-dim Gauss–Legendre quadrature rule together with a fractional operational matrix construct a constrained problem, that is solved by the Lagrange multipliers method. The convergence of the GBP method is proved and its efficiency is demonstrated by several examples.
TL;DR: In this article, a collocation method based on radial basis functions (RBFs) was proposed for solving V-O fractional Klein-Gordon equations, and the applicability of the proposed method is investigated by solving some numerical examples.
Abstract: In this paper, we study the variable-order (V-O) time fractional Klein–Gordon equation which widely appears in the various fields of engineering and mathematical physics. The numerical method which we have used for solving this equation is based on a combination of the radial basis functions (RBFs) method and finite difference scheme. In the first stage the V-O time-dependent derivative is discreticized, and then we approximate the solution by the RBFs. The main goal is to show that the collocation method based on RBFs is suitable for solving V-O fractional differential equations. The applicability of the proposed method is investigated by solving some numerical examples. The obtained results show that the proposed approach is very efficient and accurate. Also, the effect of replacing V-O fractional derivative of order
$$\alpha (x,t)$$
with its approximations on the behavior of approximate solutions relative to the exact solution is investigated numerically.
TL;DR: In this article, the Haar wavelet method for solving boundary value problems is described and the experimental results confirm the computational efficiency and simplicity of the proposed method, and also the implementation of the method for problems arising in the real world for phenomena in fluid mechanics and construction engineering approves the applicability of the approach for a variety of problems.
Abstract: In This paper, the developed Haar wavelet method for solving boundary value problems is described. As known, the orthogonal Haar basis functions are applied widely for solving initial value problems, but In this study, the method for solving systems of ODEs associated with multipoint boundary conditions is generalized in separated or non-separated forms. In this technique, a system of high-order boundary value problems of ordinary differential equations is reduced to a system of algebraic equations. The experimental results confirm the computational efficiency and simplicity of the proposed method. Also, the implementation of the method for solving the systems arising in the real world for phenomena in fluid mechanics and construction engineering approves the applicability of the approach for a variety of problems.
TL;DR: In this paper, the authors investigated simultaneous effects of pressure, temperature, impurity location, Rashba and Dresselhaus spin-orbit interaction and magnetic field on realized GaAs/Al 0.5Ga0.5As pyramid quantum dot with considering the wet layer.
Abstract: We have investigated simultaneous effects of pressure, temperature, impurity location, Rashba and Dresselhaus spin–orbit interaction and magnetic field on THG of realized GaAs/Al0.5Ga0.5As pyramid quantum dot with considering the wet layer. For this purpose, we have calculated the energy levels and wave function of one electron that is confined in constant potential, in presence of impurity, magnetic field, Rashba and Dresselhous SOI in various temperatures, pressure and impurity location in effective mass approximation by FEM. In the following, we have presented the effect of magnetic field, Rashba and Dresselhous SOI, temperature, pressure and impurity location on THG in various conditions. Results show that: (1) THG increases by increasing the magnetic field and the distance between the peaks decrease. Also, the peak corresponding to the E21 transition has a blue shift and the peaks corresponding to the E31/2 and E41/3 transitions have a red shift. (2) THG decreases and shifts to higher energies and has a small blue shift by augment of temperature. (3) THG enhances and all of peaks have a red shift by increment of pressure. (4) THG has a minimum around z0 = 4 nm in all magnetic field, temperature and pressure cases. Also, all of peaks have a blue shift by augment of z0 until z0 = 4 nm and then have a red shit by increasing the z0.
TL;DR: In this article, a Taylor's series expansion method is developed as a suitable high-precision method for solving variable-order fractional optimization problems subject to a partial differential equation.
Abstract: This article introduces a category of variable-order fractional optimization problems subject to a partial differential equation. The Taylor’s series expansion method is developed as a suitable high-precision method for solving such problems. The proposed approach through the consecutive formulation describes how an optimization problem can be transformed into a system of algebraic equations using the Taylor’s series expansion. This achievement is significantly advantageous since the obtained algebraic system straightforwardly can be solved. The accuracy of the presented method is analyzed by solving a number of numerical examples.
TL;DR: In this paper, the authors introduce a new class of optimal control problems governed by a dynamical system of weakly singular variable-order fractional integral equations and define a new set of control problems for the system.
Abstract: The main objectives of this study are to introduce a new class of optimal control problems governed by a dynamical system of weakly singular variable-order fractional integral equations and to esta...
08 Sep 2020