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Solving initial and boundary value problems of fractional ordinary differential equations by using collocation and fractional powers

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2019年 354卷 348-359页

作者： Simoes Patricio, M. F. Ramos, Higinio Patricio, Miguel Univ Coimbra Dept Matemat CMUC P-3001454 Coimbra Portugal Univ Salamanca Sci Comp Grp Salamanca Spain Escuela Politecn Super Campus Viriato Zamora 49022 Spain Univ Coimbra Fac Med Lab Biostat & Med Informat Coimbra Portugal Univ Coimbra Fac Med IBILI Coimbra Portugal Univ Coimbra CMUC Coimbra Portugal

This paper presents a novel approach for solving initial and boundary -values problems on ordinary fractional differential equations. The strategy considered is based on the use of a suitable truncated series expressed in terms of fractional powers of the independent variable, and a collocation approach. With this strategy accurate numerical approximations can be obtained. The error criterion is based on the concepts of residue and relative error. Two algorithms fitted to the nature (linear or nonlinear) of the considered fractional differential equation are proposed. Finally, some problems illustrating the efficiency of the proposed method are presented. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Caputo fractional derivative Fractional differential equations Gamma function Collocation method initial and boundary value problems

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boundary value technique based on a fifth derivative method of order 10 for fourth order initial and boundary value problems

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AFRIKA MATEMATIKA 2018年第5-6期29卷 699-718页

作者： Oladejo, H. B. Jator, S. N. Areo, E. A. FUTA Dept Math Sci Akure Nigeria Austin Peay State Univ Dept Math & Stat Clarksville TN 37044 USA

In this paper, the fourth order ordinary differential equation (ODE) of the form y(()(iv)()) = f (x, y, y', y '', y''') subject to initial or boundary conditions is solved directly without reducing the higher ODE into an equivalent first order system. This is accomplished by constructing a continuous fifth derivative method (CFIDM) which is used to produce the main and additional discrete fifth derivative methods (FIDMs). The FIDMs are then assembled and applied as a single block matrix equation to simultaneously provide all approximations on the entire interval of interest for the initial or boundary value problem. The convergence analysis of the method is discussed. The accuracy of the method is illustrated via numerical experiments.

关键词： Fourth order initial and boundary value problems Fifth derivative method Convergence

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Solving initial and boundary value problems using learning automata particle swarm optimization

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ENGINEERING OPTIMIZATION 2015年第5期47卷 656-673页

作者： Nemati, Kourosh Shamsuddin, Siti Mariyam Darus, Maslina Univ Teknol Malaysia Fac Comp UTM Big Data Ctr Johor Baharu 81310 Malaysia Univ Kebangsaan Malaysia Fac Sci & Technol Sch Math Sci Bangi 43600 Selangor Darul Malaysia

In this article, the particle swarm optimization (PSO) algorithm is modified to use the learning automata (LA) technique for solving initial and boundary value problems. A constrained problem is converted into an unconstrained problem using a penalty method to define an appropriate fitness function, which is optimized using the LA-PSO method. This method analyses a large number of candidate solutions of the unconstrained problem with the LA-PSO algorithm to minimize an error measure, which quantifies how well a candidate solution satisfies the governing ordinary differential equations (ODEs) or partial differential equations (PDEs) and the boundary conditions. This approach is very capable of solving linear and nonlinear ODEs, systems of ordinary differential equations, and linear and nonlinear PDEs. The computational efficiency and accuracy of the PSO algorithm combined with the LA technique for solving initial and boundary value problems were improved. Numerical results demonstrate the high accuracy and efficiency of the proposed method.

关键词： penalty method learning automata initial and boundary value problems unconstrained optimization particle swarm optimization

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An optimization technique based on imperialist competition algorithm to measurement of error for solving initial and boundary value problems

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MEASUREMENT 2014年第1期48卷 96-108页

作者： Nemati, K. Shamsuddin, S. M. Darus, M. Univ Teknol Malaysia Fac Comp Soft Comp Res Grp Johor Baharu 81310 Malaysia Univ Kebangsaan Malaysia Fac Sci & Technol Sch Math Sci Bangi 43600 Selangor Darul Malaysia

Imperialist competitive algorithm (ICA) is proposed to solve initial and boundary value problems in this paper. A constrained problem is converted into an unconstrained problem through the use of a penalty method in other to define an appropriate fitness function that is optimized by means of the ICA method. The methodology adopted evaluates a large number of candidate solutions of the unconstrained problem with the ICA to minimize error measure, which quantifies how well a candidate solution satisfies the governing ordinary differential equations (ODEs) or partial differential equations (PDEs) and the boundary conditions. The method is proficient approach to solve linear and nonlinear ODEs, systems of ordinary differential equations (SODEs), linear and nonlinear PDEs. Numerical experiments demonstrate the accuracy and efficiency of the proposed method. Thus, this method is a promising tool for solving higher-dimensional problems. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Imperialist competitive algorithm Evolutionary algorithm initial and boundary value problems Unconstrained optimization Penalty method

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Generalized separation of variable methods with their comparison: exact solutions of time-fractional nonlinear PDEs in higher dimensions

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FRACTIONAL CALCULUS AND APPLIED ANALYSIS 2024年第6期27卷 3240-3290页

作者： Prakash, P. Priyendhu, K. S. Sahadevan, R. Amrita Vishwa Vidyapeetham Amrita Sch Phys Sci Dept Math Coimbatore India Univ Madras Ramanujan Inst Adv Study Math Chennai 600005 India

A systematic investigation of the significance and applicability of two different approaches of generalized separation of variable (GSV) methods for time-fractional nonlinear PDEs in (2+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+1)$$\end{document} and (m+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m+1)$$\end{document}-dimensions is presented. Also, this work explicitly shows that while constructing the exact solutions of time-fractional nonlinear PDEs in (2+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+1)$$\end{document} and (m+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m+1)$$\end{document}-dimensions without and with delay terms, how to overcome unusual (non-standard) properties of singular kernel fractional derivatives such as chain rule, semigroup property, and the Leibniz rule. Moreover, the importance and effectiveness of the two GSV methods have been discussed through the initial and boundary value problems of the time-fractional nonlinear generalized convection-diffusion equation in (2+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$

关键词： Generalized separation of variable methods Invariant subspaces Exact solutions fractional nonlinear convection-diffusion equation initial and boundary value problems Laplace transformation technique

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NONCONVEX COERCIVE HAMILTON-JACOBI EQUATIONS: GUERAND'S RELAXATION REVISITED

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PURE AND APPLIED ANALYSIS 2024年第4期6卷 1055-1089页

作者： Forcadel, Nicolas Imbert, Cyril Monneau, Regis Inst Natl Sci Appl Rouen Rouen France Ecole Normale Super Dept Math & Applicat Paris France Univ Paris Est CERM ENPC Paris France

This work is concerned with Hamilton-Jacobi equations of evolution type posed in domains and supplemented with boundary conditions. Hamiltonians are coercive but are neither convex nor quasiconvex. We analyze boundary conditions when understood in the sense of viscosity solutions. This analysis is based on the study of boundary conditions of evolution type. More precisely, we give a new formula for the relaxed boundary conditions derived by J. Guerand (J. Differential Equations 263 :5 (2017), 2812-2850). This new point of view unveils a connection between the relaxation operator and the classical Godunov flux from the theory of conservation laws. We apply our methods to two classical boundary value problems. It is shown that the relaxed Neumann boundary condition is expressed in terms of Godunov's flux while the relaxed Dirichlet boundary condition reduces to an obstacle problem at the boundary associated with the lower nonincreasing envelope of the Hamiltonian.

关键词： Hamilton-Jacobi equations initial and boundary value problems viscosity solutions

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Nonlinear two-component system of time-fractional PDEs in (2+1)- dimensions: Invariant subspace method combined with variable transformation

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2024年 137卷

作者： Prakash, P. Priyendhu, K. S. Lakshmanan, M. Amrita Vishwa Vidyapeetham Amrita Sch Phys Sci Dept Math Coimbatore Tamil Nadu India Bharathidasan Univ Dept Nonlinear Dynam Tiruchirapalli 620024 India

In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two -component system of time -fractional PDEs (TFPDEs) in (2 + 1) - dimensions for the first time. Also, we explicitly explain how to construct various kinds of finite -dimensional invariant linear product spaces for the given system using the invariant subspace method combined with variable transformation. Additionally, we present how to use the obtained invariant linear product spaces to derive the generalized separable exact solutions of the discussed system. We also note that the discussed method will help to reduce the nonlinear two -component system of TFPDEs in (2 + 1) - dimensions into the nonlinear two -component system of TFPDEs in (1+1)dimensions, which again reduces to a system of time -fractional ODEs through the obtained invariant linear product spaces. More specifically, the significance and efficacy of the systematic investigation of the discussed method have been investigated through the initial and boundary value problems of the generalized nonlinear two -component system of time -fractional reaction- diffusion equations (TFRDEs) in (2 + 1) - dimensions for finding the generalized separable exact solutions, which can be expressed in terms of the exponential, trigonometric, polynomial, EulerGamma and Mittag-Leffler functions. Also, 2D and 3D graphical representations of some of the obtained solutions are presented for different values of fractional orders.

关键词： Fractional reaction-diffusion system Invariant subspace method associated with variable transformation initial and boundary value problems System of nonlinear PDEs Exact solutions Laplace transformation technique

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Numerical simulation of discretized second-order variable coefficient elliptic PDEs by a Classical Eight-step Model

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RESULTS IN PHYSICS 2022年第0期41卷

作者： Adeyefa, Emmanuel Oluseye Omole, Ezekiel Olaoluwa Shokri, Ali Nonlaopon, Kamsing Fed Univ Oye Ekiti Fac Sci Dept Math PMB 373 Oye Ekiti Ekiti Nigeria Univ Maragheh Fac Math Sci Maragheh Iran Khon Kaen Univ Dept Math Khon Kaen 40002 Thailand

This article presents a new numerical model with nodes within the interval of eight-step for numerical simulation of a class of variable coefficient elliptic partial differential equations in the two-dimensional domain. The method is developed via the principle of collocation and interpolation techniques using Hermite polynomial as the basis function. The main classical model and its derivatives generated from the continuous function are then united together to form the required Classical Eight-step Model (CEM). The analysis of the CEM was investigated and shows that it satisfies the conditions for convergence with an algebraic order of nine. The CEM is applied to solve the semi-discretized elliptic partial differential equations with a variable coefficient arising from the discretization of one of their spatial variables. The performance of the CEM was established with six test problems. The approximate solution generated using CEM is compared with the analytical solution of the problems and other existing methods in the literature. Numerical illustrations demonstrate that the method is convergent and highly accurate. The advantages of CEM over other existing methods reveal the accuracy and efficiency of the CEM as depicted in curves.

关键词： Classical Eight-step Model Convergence analysis Discretization Helmholtz equation Method of Lines Laplace equation Variable coefficient Elliptic PDEs Poisson equation initial and boundary value problems

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Tau and Galerkin operational matrices of derivatives for treating singular and Emden-Fowler third-order-type equations

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INTERNATIONAL JOURNAL OF MODERN PHYSICS C 2022年第5期33卷 2250061-2250061页

作者： Abd-Elhameed, W. M. Ahmed, H. M. Cairo Univ Fac Sci Dept Math Giza 12613 Egypt Helwan Univ Fac Technol & Educ Dept Math Cairo Egypt

In this paper, our target is to implement and analyze numerical algorithms for the numerical solutions of initial and boundary third-order singular-type equations, and in particular the Emden-Fowler-type equations. Both linear and nonlinear equations are investigated. Two kinds of basis functions in terms of the shifted Chebyshev polynomials of the second kind are employed. Regarding the initial value problems, a new Galerkin operational matrix of derivatives is established and utilized, while in the case of boundary value problems (BVPs), the operational matrix of derivatives of the shifted Chebyshev polynomials of the second kind is employed. Convergence and error estimates of the two proposed expansions are given. The obtained solutions are spectral and they are consequences of the application of tau and collocation methods. These methods serve to convert the problems governed by their initial/boundary conditions into systems of linear or nonlinear algebraic equations which can be solved via suitable numerical solvers. Finally, we support our theoretical study by presenting several examples to ensure the accuracy and efficiency of the proposed algorithms.

关键词： Shifted second-kind Chebyshev polynomials initial and boundary value problems tau and collocation algorithms Emden-Fowler-type equations

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Ideal Gas Outflow from a Cylindrical or Spherical Source into a Vacuum

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FLUID DYNAMICS 2018年第5期53卷 596-607页

作者： Valiev, Kh. F. Kraiko, A. N. Baranov Cent Inst Aviat Motors CIAM Ul Aviamotornaya 2 Moscow 111116 Russia Moscow Inst Sci & Technol Inst Per 9 Dolgoprudnyi 141701 Moscow Oblast Russia

The solutions of initial and boundary value problems of the outflow of an ideal (inviscid and non-heat-conducting) gas from cylindrical and spherical sources into a vacuum are obtained. Time is measured from the moment, when the source is turned on;at this moment the source is surrounded by a vacuum. The entropy, flow rate, and the Mach number of the gas outflowing from the source are given, together with the source radius;the Mach number can be greater of or equal to unity. If the source radius is greater than zero, then the flow domain in the radial coordinate-time plane consists of the stationary source flow and adjoining non-self-similar centered expansion wave consisting of C--characteristics. The stationary flow is described by the known formulas, while the expansion wave is calculated by the method of characteristics. The calculations by this method confirm the earlier obtained laws for large values of the radial coordinate. The interface between the vacuum and the expansion wave is the straight trajectory of particles and, at the same time, a unique rectilinear C--characteristic. For the source of zero radius (pointwise source) the velocity, density, and speed of sound of the outflowing gas are infinite. The gas velocity remains infinite everywhere, while the density and speed of sound become zero for any non-zero values of the radial coordinate. For the pointwise source the problem of outflow into a vacuum is self-similar. In the plane of the self-similar velocity and speed of sound its solution is given by three singular points of a differential equation in these variables. At one of these points the self-similar velocity is infinite, the self-similar speed of sound is zero, and the self-similar independent variable varies from zero to infinity, with the exception of the extreme values.

关键词： initial and boundary value problems cylindrical and spherical sources outflow into a vacuum flow structures for nonzero and zero source radii self-similar solution for the pointwise source

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