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Shifted Boubaker Lagrangian approach for solving biological systems

International Journal of Biomathematics 2018年第3期11卷 155-175页

作者： Kourosh Parand Hossein Yousefi Mina Fotouhifar mehdi delkhosh mehdi Hosseinzadeh Department of Computer Sciences Shahid Beheshti University G.C. Tehran Iran Department of Cognitive Modelling Institute for Cognitive and Brain Sciences Shahid Beheshti University G. C. Tehran Iran Department of Computer Sciences Shahid Beheshti University G. C. Tehran Iran Iran University of Medical Sciences Tehran Iran Computer Science University of Human Development Sulaimaniyah Iraq

Mathematical models and computer simulations are useful experimental tools for building and testing theories. Many mathematical models in biology can be formulated by a nonlinear system of ordinary differential equations. This work deals with the numerical solution of the hantavirus infection model, the human immunodeficiency virus （HIV） infection model of CD4^＋T cells and the susceptible-infected-removed （SIR） epidemic model using a new reliable algorithm based on shifted Boubaker Lagrangian （SBL） method. This method reduces the solution of such system to a system of linear or non- linear algebraic equations which are solved using the Newton iteration method. The obtained results of the proposed method show highly accurate and valid for an arbitrary finite interval. Also, those are compared with fourth-order Runge-Kutta （RK4） method and with the solutions obtained by some other methods in the literature.

关键词： Hantavirus infection model HIV infection model of CD4^+T cells SIR epi- demic model system of nonlinear differential equations Lagrangian method Boubaker polynomials.

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An Efficient Numerical Solution of Nonlinear Hunter-Saxton Equation

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Communications in Theoretical Physics 2017年第5期67卷 483-492页

作者： Kourosh Parand mehdi delkhosh Department of Computer Sciences Shahid Beheshti UniversityG.C.TehranIran Department of Cognitive Modelling Institute for Cognitive and Brain SciencesShahid Beheshti UniversityG.C.TehranIran

In this paper, the nonlinear Hunter–Saxton equation, which is a famous partial differential equation,is solved by using a hybrid numerical method based on the quasilinearization method and the bivariate generalized fractional order of the Chebyshev functions(B-GFCF) collocation method. First, using the quasilinearization method,the equation is converted into a sequence of linear partial differential equations(LPD), and then these LPDs are solved using the B-GFCF collocation method. A very good approximation of solutions is obtained, and comparisons show that the obtained results are more accurate than the results of other researchers.

关键词： Hunter–Saxton equation fractional order of the Chebyshev functions quasilinearization method collocation method nonlinear PDE

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Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) Collocation Method for Solving Linear and Nonlinear Fokker-Planck Equations

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Communications in Theoretical Physics 2018年第5期69卷 519-531页

作者： K.Parand S.Latifi M.M.Moayeri M.delkhosh Department of Computer Sciences Shahid Beheshti University Department of Cognitive Modeling Institute for Cognitive and Brain SciencesShahid Beheshti University

In this study, we have constructed a new numerical approach for solving the time-dependent linear and nonlinear Fokker-Planck equations. In fact, we have discretized the time variable with Crank-Nicolson method and for the space variable, a numerical method based on Generalized Lagrange Jacobi Gauss-Lobatto(GLJGL) collocation method is applied. It leads to in solving the equation in a series of time steps and at each time step, the problem is reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. One can observe that the proposed method is simple and accurate. Indeed, one of its merits is that it is derivative-free and by proposing a formula for derivative matrices, the difficulty aroused in calculation is overcome, along with that it does not need to calculate the General Lagrange basis and matrices; they have Kronecker property. Linear and nonlinear Fokker-Planck equations are given as examples and the results amply demonstrate that the presented method is very valid, effective,reliable and does not require any restrictive assumptions for nonlinear terms.

关键词： Fokker-Planck equations Generalized Lagrange functions Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) collocation Crank-Nicolson technique

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A Computationally Hybrid Method for Solving a Famous Physical Problem on an Unbounded Domain

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Communications in Theoretical Physics 2019年第1期71卷 9-15页

作者： F.A.Parand Z.Kalantari M.delkhosh F.Mirahmadian Department of Mathematics and Computer Science Allameh Tabataba'i University Department of Computer Sciences Shahid Beheshti University Department of Mathematics and Computer Science Islamic Azad UniversityBardaskan Branch

In this paper, a hybrid method based on the collocation and Newton-Kantorovich methods is used for solving the nonlinear singular Thomas-Fermi equation. At first, by using the Newton-Kantorovich method, the nonlinear problem is converted to a sequence of linear differential equations, and then, the fractional order of rational Legendre functions are introduced and used for solving linear differential equations at each iteration based on the collocation method. Moreover, the boundary conditions of the problem by using Ritz method without domain truncation method are satisfied. In the end, the obtained results compare with other published in the literature to show the performance of the method, and the amounts of residual error are very small, which indicates the convergence of the method.

关键词： fractional order of rational Legendre functions Newton-Kantorovich method collocation method Thomas-Fermi equation

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