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Numerical Methods for a Class of Hybrid Weakly Singular Integro-Differential Equations 被引量:1
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作者 Shihchung Chiang 《Applied Mathematics》 2017年第7期956-966,共11页
This paper proposes numerical methods for solving hybrid weakly singular integro-differential equations of the second kind. The terms in these equations are in the following order: derivative term of a state, integro-... This paper proposes numerical methods for solving hybrid weakly singular integro-differential equations of the second kind. The terms in these equations are in the following order: derivative term of a state, integro-differential term of a state with a weakly singular kernel, a state, integral term of a state with a smooth kernel, and force. The original class of weakly singular integro-differential equations of the first kind is derived from aeroelasticity mathematical models. Among the proposed methods, the method for solving linear cases is fully based on previously reported approximation scheme for equations of the first kind. For nonlinear cases, a revised method is proposed. Examples are presented to demonstrate the effectiveness of the proposed methods, and the results indicate that the proposed methods facilitate achieving satisfactory and accurate approximations. 展开更多
关键词 HYBRID weakly singular integro-differential equations
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Optimization in Transition between Two Dynamic Systems Governed by a Class of Weakly Singular Integro-Differential Equations
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作者 Shihchung Chiang 《Applied Mathematics》 2019年第10期826-835,共10页
This study presents numerical methods for solving the minimum energies that satisfy typical optimal requirements in the transition between two dynamic systems where each system is governed by a different kind of weakl... This study presents numerical methods for solving the minimum energies that satisfy typical optimal requirements in the transition between two dynamic systems where each system is governed by a different kind of weakly singular integro-differential equation. The class of weakly singular integro-differential equations originates from mathematical models in aeroelasticity. The proposed numerical methods are based on earlier reported approximation schemes for the equations of the first kind and the second kind. The main result of this study is the development of numerical techniques for determining the stability between two dynamic systems in the minimum energy sense. 展开更多
关键词 Optimal REQUIREMENT TRANSITION weakly singular integro-differential equations Stability
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On the Well-Posedness of a Class of Hybrid Weakly Singular Integro-Differential Equations
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作者 Shihchung Chiang 《Applied Mathematics》 2022年第10期793-798,共6页
In this study, a revised version of some numerical methods for a class of hybrid integro-differential equations with weakly singular kernels (Abel types) is presented. These equations were developed from a class of in... In this study, a revised version of some numerical methods for a class of hybrid integro-differential equations with weakly singular kernels (Abel types) is presented. These equations were developed from a class of integro-differential equations of first kind originating from an aeroelasticity problem. By manipulating the bounds of initial conditions with random variations, this study numerically demonstrated the well-posedness properties of the equations. Finally, an assumption of separating variables, allowed for linear splines to be chosen as a basis and for the differentiation and integration of the integro-differential part to be interchanged;hence, a numerical scheme was constructed. 展开更多
关键词 WELL-POSEDNESS HYBRID weakly singular integro-differential equations
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A Differential Quadrature Based Approach for Volterra Partial Integro-Differential Equation with a Weakly Singular Kernel 被引量:1
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作者 Siraj-ul-Islam Arshed Ali +1 位作者 Aqib Zafar Iltaf Hussain 《Computer Modeling in Engineering & Sciences》 SCIE EI 2020年第9期915-935,共21页
Differential quadrature method is employed by numerous researchers due to its numerical accuracy and computational efficiency,and is mentioned as potential alternative of conventional numerical methods.In this paper,a... Differential quadrature method is employed by numerous researchers due to its numerical accuracy and computational efficiency,and is mentioned as potential alternative of conventional numerical methods.In this paper,a differential quadrature based numerical scheme is developed for solving volterra partial integro-differential equation of second order having a weakly singular kernel.The scheme uses cubic trigonometric B-spline functions to determine the weighting coefficients in the differential quadrature approximation of the second order spatial derivative.The advantage of this approximation is that it reduces the problem to a first order time dependent integro-differential equation(IDE).The proposed scheme is obtained in the form of an algebraic system by reducing the time dependent IDE through unconditionally stable Euler backward method as time integrator.The scheme is validated using a homogeneous and two nonhomogeneous test problems.Conditioning of the system matrix and numerical convergence of the method are analyzed for spatial and temporal domain discretization parameters.Comparison of results of the present approach with Sinc collocation method and quasi-wavelet method are also made. 展开更多
关键词 Partial integro-differential equation differential quadrature cubic trigonometric B-spline functions weakly singular kernel
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A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY
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作者 Weishan ZHENG Yanping CHEN 《Acta Mathematica Scientia》 SCIE CSCD 2022年第1期387-402,共16页
In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transforma... In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1,1],so that the Jacobi orthogonal polynomial theory can be applied conveniently.In order to obtain high order accuracy for the approximation,the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules.In the end,we provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in both the L^(∞)-norm and the weighted L^(2)-norm. 展开更多
关键词 Volterra integro-differential equation pantograph delay weakly singular kernel Jacobi-collocation spectral methods error analysis convergence analysis
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GENERALIZED JACOBI SPECTRAL GALERKIN METHOD FOR FRACTIONAL-ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH WEAKLY SINGULAR KERNELS
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作者 Yanping Chen Zhenrong Chen Yunqing Huang 《Journal of Computational Mathematics》 SCIE CSCD 2024年第2期355-371,共17页
For fractional Volterra integro-differential equations(FVIDEs)with weakly singular kernels,this paper proposes a generalized Jacobi spectral Galerkin method.The basis functions for the provided method are selected gen... For fractional Volterra integro-differential equations(FVIDEs)with weakly singular kernels,this paper proposes a generalized Jacobi spectral Galerkin method.The basis functions for the provided method are selected generalized Jacobi functions(GJFs),which can be utilized as natural basis functions of spectral methods for weakly singular FVIDEs when appropriately constructed.The developed method's spectral rate of convergence is determined using the L^(∞)-norm and the weighted L^(2)-norm.Numerical results indicate the usefulness of the proposed method. 展开更多
关键词 Generalized Jacobi spectral Galerkin method Fractional-order Volterra integ-ro-differential equations weakly singular kernels Convergence analysis
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MULTIPLE POSITIVE SOLUTIONS FOR FIRST ORDER IMPULSIVE SINGULAR INTEGRO-DIFFERENTIAL EQUATIONS ON THE HALF LINE 被引量:7
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作者 郭大钧 《Acta Mathematica Scientia》 SCIE CSCD 2012年第6期2176-2190,共15页
In this paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive singular integro-differential equations on the half line by means of the fixed point ... In this paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive singular integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type. 展开更多
关键词 impulsive singular integro-differential equation infinite boundary valueproblem fixed point theorem of cone expansion and compression with normtype
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Wavelet Numerical Solutions for Weakly Singular Fredholm Integral Equations of the Second Kind 被引量:1
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作者 TANG Xinjian PANG Zhicheng +1 位作者 ZHU Tonglin LIU Jian 《Wuhan University Journal of Natural Sciences》 CAS 2007年第3期437-441,共5页
Daubechies interval cally weakly singular Fredholm kind. Utilizing the orthogonality equation is reduced into a linear wavelet is used to solve nurneriintegral equations of the second of the wavelet basis, the integra... Daubechies interval cally weakly singular Fredholm kind. Utilizing the orthogonality equation is reduced into a linear wavelet is used to solve nurneriintegral equations of the second of the wavelet basis, the integral system of equations. The vanishing moments of the wavelet make the wavelet coefficient matrices sparse, while the continuity of the derivative functions of basis overcomes naturally the singular problem of the integral solution. The uniform convergence of the approximate solution by the wavelet method is proved and the error bound is given. Finally, numerical example is presented to show the application of the wavelet method. 展开更多
关键词 weakly singular integral equations interval wavelet sparse matrix
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Numerical Minimum Time Control to a Class of Singular Integro-Differential Equations
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作者 Shihchung Chiang 《Journal of Mathematics and System Science》 2015年第2期72-82,共11页
This study presents a numerical method for determining the minimum time required for the states of one class of integro-differential equations of the first kind to reach its attainable region by assuming the forcing t... This study presents a numerical method for determining the minimum time required for the states of one class of integro-differential equations of the first kind to reach its attainable region by assuming the forcing terms of the equations as controls. These equations consist of integro-differential parts containing weakly singular kernels. The feasibility of the numerical method is demonstrated by comparing the minimum time and corresponding possible time by using extreme controls to reach the attainable region under different initial conditions. 展开更多
关键词 integro-differential equations weakly singular kernels minimum time control
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EXISTENCE AND UNIQUENESS RESULTS FOR BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS INVOLVING GRONWALL'S INEQUALITY IN BANACH SPACES 被引量:2
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作者 Dimplekumar N. CHALISHAJAR K. KARTHIKEYAN 《Acta Mathematica Scientia》 SCIE CSCD 2013年第3期758-772,共15页
We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ (n-1, n) in Banach spaces. Existence and uniqueness results of solutions are established by vi... We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ (n-1, n) in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder's inequality, a suitable singular Cronwall's inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results. 展开更多
关键词 Boundary value problems fractional order integro-differential equations bound-ary value problems existence and uniqueness singular gronwall inequality fixed point theorem
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Numerical Algorithms for Solving One Type of Singular Integro-Differential Equation Containing Derivatives of the Time Delay States 被引量:1
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作者 Shihchung Chiang Terry L. Herdman 《Applied Mathematics》 2015年第8期1294-1301,共8页
This study presents numerical algorithms for solving a class of equations that partly consists of derivatives of the unknown state at previous certain times, as well as an integro-differential term containing a weakly... This study presents numerical algorithms for solving a class of equations that partly consists of derivatives of the unknown state at previous certain times, as well as an integro-differential term containing a weakly singular kernel. These equations are types of integro-differential equation of the second kind and were originally obtained from an aeroelasticity problem. One of the main contributions of this study is to propose numerical algorithms that do not involve transforming the original equation into the corresponding Volterra equation, but still enable the numerical solution of the original equation to be determined. The feasibility of the proposed numerical algorithm is demonstrated by applying examples in measuring the maximum errors with exact solutions at every computed nodes and calculating the corresponding numerical rates of convergence thereafter. 展开更多
关键词 integro-differential equation of the Second KIND weakly singular KERNEL Numerical Algorithms Rates of Convergence
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A Uniformly Convergent Numerical Method Using Weak Formulation for Singularly Perturbed Differential Equations
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作者 Weiqun Zhang 《Journal of Mathematics and System Science》 2019年第1期1-6,共6页
A numerical method using weak formulation is proposed to solve singularly perturbed differential equations. The numerical method is applied to both linear and nonlinear perturbation problems. A linear differential equ... A numerical method using weak formulation is proposed to solve singularly perturbed differential equations. The numerical method is applied to both linear and nonlinear perturbation problems. A linear differential equation is solved using its weak formulation with a test space composed of exponential functions matching boundary layers. A nonlinear singular perturbation problem is converted into a system of linear differentiation equations. Then each linear differential equation is solved iteratively. The uniform convergence, which is independent of the singular perturbation parameter, is numerically verified. 展开更多
关键词 singular PERTURBATION differential equations boundary layers numerical methods weak formulation
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A Comparative Numerical Study of Parabolic Partial Integro-Differential Equation Arising from Convection-Diffusion
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作者 Kamil Khan Arshed Ali +2 位作者 Fazal-i-Haq Iltaf Hussain Nudrat Amir 《Computer Modeling in Engineering & Sciences》 SCIE EI 2021年第2期673-692,共20页
This article studies the development of two numerical techniques for solving convection-diffusion type partial integro-differential equation(PIDE)with a weakly singular kernel.Cubic trigonometric B-spline(CTBS)functio... This article studies the development of two numerical techniques for solving convection-diffusion type partial integro-differential equation(PIDE)with a weakly singular kernel.Cubic trigonometric B-spline(CTBS)functions are used for interpolation in both methods.The first method is CTBS based collocation method which reduces the PIDE to an algebraic tridiagonal system of linear equations.The other method is CTBS based differential quadrature method which converts the PIDE to a system of ODEs by computing spatial derivatives as weighted sum of function values.An efficient tridiagonal solver is used for the solution of the linear system obtained in the first method as well as for determination of weighting coefficients in the second method.An explicit scheme is employed as time integrator to solve the system of ODEs obtained in the second method.The methods are tested with three nonhomogeneous problems for their validation.Stability,computational efficiency and numerical convergence of the methods are analyzed.Comparison of errors in approximations produced by the present methods versus different values of discretization parameters and convection-diffusion coefficients are made.Convection and diffusion dominant cases are discussed in terms of Peclet number.The results are also compared with cubic B-spline collocation method. 展开更多
关键词 Partial integro-differential equation CONVECTION-DIFFUSION collocation method differential quadrature cubic trigonometric B-spline functions weakly singular kernel
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FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES WITH SINGULAR INITIAL DATA L^p(P<∞)
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作者 赵会江 《Acta Mathematica Scientia》 SCIE CSCD 1996年第3期308-320,共13页
We study the existence problem for the equations of first order quasilinearequations in several inpendent variables with singular initial data Lp(P<∞). We the convergence of the Lp(P<∞) bounded approximating s... We study the existence problem for the equations of first order quasilinearequations in several inpendent variables with singular initial data Lp(P<∞). We the convergence of the Lp(P<∞) bounded approximating sequences generatedby the method of vanishing viscosity. The uniqueness of the generalized solutions whichcan be obtained by the method of vanishing viscosity is also obtained. 展开更多
关键词 singular initial data quasilinear equations global weak solutions
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Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions 被引量:4
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作者 Yunxia Wei Yanping Chen 《Advances in Applied Mathematics and Mechanics》 SCIE 2012年第1期1-20,共20页
The theory of a class of spectral methods is extended to Volterra integrodifferential equations which contain a weakly singular kernel(t−s)^(−μ) with 0<μ<1.In this work,we consider the case when the underlying... The theory of a class of spectral methods is extended to Volterra integrodifferential equations which contain a weakly singular kernel(t−s)^(−μ) with 0<μ<1.In this work,we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth.We provide a rigorous error analysis for the spectral methods,which shows that both the errors of approximate solutions and the errors of approximate derivatives of the solutions decay exponentially in L^(∞)-norm and weighted L^(2)-norm.The numerical examples are given to illustrate the theoretical results. 展开更多
关键词 ∞Volterra integro-differential equations weakly singular kernels spectral methods convergence analysis
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SOME RESULTS ON A DEGENERATE AND SINGULAR DIFFUSION EQUATION 被引量:1
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作者 姚正安 周文书 《Acta Mathematica Scientia》 SCIE CSCD 2007年第3期581-601,共21页
In this article, a degenerate and singular diffusion problem is studied. The existence of solutions is established by parabolic regularization. Some properties of solutions, for instance, asymptotic behavior, are also... In this article, a degenerate and singular diffusion problem is studied. The existence of solutions is established by parabolic regularization. Some properties of solutions, for instance, asymptotic behavior, are also discussed. 展开更多
关键词 Degenerate and singular diffusion equation weak solution EXISTENCE asymptotic behavior
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A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel 被引量:1
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作者 Yunxia Wei Yanping Chen 《Numerical Mathematics(Theory,Methods and Applications)》 SCIE 2013年第2期424-446,共23页
This paper is concerned with obtaining an approximate solution and an approximate derivative of the solution for neutral Volterra integro-differential equation with a weakly singular kernel.The solution of this equati... This paper is concerned with obtaining an approximate solution and an approximate derivative of the solution for neutral Volterra integro-differential equation with a weakly singular kernel.The solution of this equation,even for analytic data,is not smooth on the entire interval of integration.The Jacobi collocation discretization is proposed for the given equation.A rigorous analysis of error bound is also provided which theoretically justifies that both the error of approximate solution and the error of approximate derivative of the solution decay exponentially in L∞norm and weighted L2 norm.Numerical results are presented to demonstrate the effectiveness of the spectral method. 展开更多
关键词 Neutral Volterra integro-differential equation weakly singular kernel Jacobi collocation discretization convergence analysis
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The Global Behavior of Finite Difference-Spatial Spectral Collocation Methods for a Partial Integro-differential Equation with a Weakly Singular Kernel
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作者 Jie Tang Da Xu 《Numerical Mathematics(Theory,Methods and Applications)》 SCIE 2013年第3期556-570,共15页
The z-transform is introduced to analyze a full discretization method fora partial integro-differential equation (PIDE) with a weakly singular kernel. In thismethod, spectral collocation is used for the spatial discre... The z-transform is introduced to analyze a full discretization method fora partial integro-differential equation (PIDE) with a weakly singular kernel. In thismethod, spectral collocation is used for the spatial discretization, and, for the time stepping, the finite difference method combined with the convolution quadrature rule isconsidered. The global stability and convergence properties of complete discretizationare derived and numerical experiments are reported. 展开更多
关键词 Partial integro-differential equation weakly singular kernel spectral collocation methods Z-TRANSFORM convolution quadrature
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A Pseudo-Spectral Scheme for Systems of Two-Point Boundary Value Problems with Left and Right Sided Fractional Derivatives and Related Integral Equations
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作者 I.G.Ameen N.A.Elkot +2 位作者 M.A.Zaky A.S.Hendy E.H.Doha 《Computer Modeling in Engineering & Sciences》 SCIE EI 2021年第7期21-41,共21页
We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left-and right-sided fractional derivatives.The main ingredient of the proposed method is to recast the p... We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left-and right-sided fractional derivatives.The main ingredient of the proposed method is to recast the problem into an equivalent system of weakly singular integral equations.Then,a Legendre-based spectral collocation method is developed for solving the transformed system.Therefore,we can make good use of the advantages of the Gauss quadrature rule.We present the construction and analysis of the collocation method.These results can be indirectly applied to solve fractional optimal control problems by considering the corresponding Euler–Lagrange equations.Two numerical examples are given to confirm the convergence analysis and robustness of the scheme. 展开更多
关键词 Spectral collocation method weakly singular integral equations two-point boundary value problems convergence analysis
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SUPERGEOMETRIC CONVERGENCE OF SPECTRAL COLLOCATION METHODS FOR WEAKLY SINGULAR VOLTERRA AND FREDHOLM INTEGRAL EQUATIONS WITH SMOOTH SOLUTIONS 被引量:4
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作者 Can Huang Tao Tang Zhimin Zhang 《Journal of Computational Mathematics》 SCIE CSCD 2011年第6期698-719,共22页
A spectral collocation method is proposed to solve Volterra or Fredholm integral equations with weakly singular kernels and corresponding integro-differential equations by virtue of some identities. For a class of fun... A spectral collocation method is proposed to solve Volterra or Fredholm integral equations with weakly singular kernels and corresponding integro-differential equations by virtue of some identities. For a class of functions that satisfy certain regularity conditions on a bounded domain, we obtain geometric or supergeometric convergence rate for both types of equations. Numerical results confirm our theoretical analysis. 展开更多
关键词 weakly singular kernel integro-differential equations Collocation method.
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