共查询到20条相似文献,搜索用时 15 毫秒
1.
Xiulian Shi Yunxia Wei Fenglin Huang 《Numerical Methods for Partial Differential Equations》2019,35(2):576-596
In this paper, a spectral collocation approximation is proposed for neutral and nonlinear weakly singular Volterra integro‐differential equations (VIDEs) with non‐smooth solutions. We use some suitable variable transformations to change the original equation into a new equation, so that the solution of the resulting equation possesses better regularity, and the the Jacobi orthogonal polynomial theory can be applied conveniently. Under reasonable assumptions on the nonlinearity, we carry out a rigorous error analysis in L∞ norm and weighted L2 norm. To perform the numerical simulations, some test examples (linear and nonlinear) are considered with nonsmooth solutions, and numerical results are presented. Further more, the comparative study of the proposed methods with some existing numerical methods is provided. 相似文献
2.
Farshid Mirzaee Sahar Alipour 《Numerical Methods for Partial Differential Equations》2019,35(3):1134-1151
In this paper, an effective numerical approach based on a new two‐dimensional hybrid of parabolic and block‐pulse functions (2D‐PBPFs) is presented for solving nonlinear partial quadratic integro‐differential equations of fractional order. Our approach is based on 2D‐PBPFs operational matrix method together with the fractional integral operator, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved, and the solution of fractional nonlinear partial quadratic integro‐differential equations is achieved. Convergence analysis and an error estimate associated with the proposed method is obtained, and it is proved that the numerical convergence order of the suggested numerical method is O(h3) . The validity and applicability of the method are demonstrated by solving three numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the exact solutions much easier. 相似文献
3.
Najeeb Alam Khan Asmat Ara Muhammad Jamil 《Mathematical Methods in the Applied Sciences》2011,34(14):1733-1738
In this paper, we apply the new homotopy perturbation method to solve the Volterra's model for population growth of a species in a closed system. This technique is extended to give solution for nonlinear integro‐differential equation in which the integral term represents the total metabolism accumulated fromtime zero. The approximate analytical procedure only depends on two components. The newhomotopy perturbationmethodwas applied to nonlinear integro‐differential equations directly and by converting the problem into nonlinear ordinary differential equation. We also compare this method with some other numerical results and show that the present approach is less computational and is applicable for solving nonlinear integro‐differential equations and ordinary differential equations as well. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
4.
Héctor Chang Lara Gonzalo Dávila 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2012
We study the regularity for solutions of fully nonlinear integro differential equations with respect to nonsymmetric kernels. More precisely, we assume that our operator is elliptic with respect to a family of integro differential linear operators where the symmetric parts of the kernels have a fixed homogeneity σ and the skew symmetric parts have strictly smaller homogeneity τ . We prove a weak ABP estimate and C1,α regularity. Our estimates remain uniform as we take σ→2 and τ→1 so that this extends the regularity theory for elliptic differential equations with dependence on the gradient. 相似文献
5.
Sascha Trostorff 《Mathematical Methods in the Applied Sciences》2015,38(5):834-850
We study integro‐differential inclusions in Hilbert spaces with operator‐valued kernels and give sufficient conditions for the well‐posedness. We show that several types of integro‐differential equations and inclusions are covered by the class of evolutionary inclusions, and we therefore give criteria for the well‐posedness within this framework. As an example, we apply our results to the equations of visco‐elasticity and to a class of nonlinear integro‐differential inclusions describing phase transition phenomena in materials with memory. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
6.
Haitao Che Zhaojie Zhou Ziwen Jiang Yiju Wang 《Numerical Methods for Partial Differential Equations》2013,29(3):799-817
H1‐Galerkin mixed finite element method combined with expanded mixed element method is discussed for nonlinear pseudo‐parabolic integro‐differential equations. We conduct theoretical analysis to study the existence and uniqueness of numerical solutions to the discrete scheme. A priori error estimates are derived for the unknown function, gradient function, and flux. Numerical example is presented to illustrate the effectiveness of the proposed scheme. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
7.
Mehrdad Lakestani Mahmood Jokar Mehdi Dehghan 《Mathematical Methods in the Applied Sciences》2011,34(11):1317-1329
The main aim of this paper is to apply the trigonometric wavelets for the solution of the Fredholm integro‐differential equations of nth‐order. The operational matrices of derivative for trigonometric scaling functions and wavelets are presented and are utilized to reduce the solution of the Fredholm integro‐differential equations to the solution of algebraic equations. Furthermore, we get an estimation of error bound for this method. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
8.
Luoping Chen Yanping Chen Yunqing Huang 《Numerical Methods for Partial Differential Equations》2019,35(5):1676-1693
In this paper, we will investigate a two grid finite element discretization method for the semi‐linear hyperbolic integro‐differential equations by piecewise continuous finite element method. In order to deal with the semi‐linearity of the model, we use the two grid technique and derive that once the coarse and fine mesh sizes H, h satisfy the relation h = H2 for the two‐step two grid discretization method, the two grid method achieves the same convergence accuracy as the ordinary finite element method. Both theoretical analysis and numerical experiments are given to verify the results. 相似文献
9.
Hang Xu 《Communications in Nonlinear Science & Numerical Simulation》2009,14(5):1978-1983
In this paper, we apply the homotopy analysis method (HAM) to solve the fractional Volterra’s model for population growth of a species in a closed system. This technique is extended to give solutions for nonlinear fractional integro–differential equations. The whole HAM solution procedure for nonlinear fractional differential equations is established. Further, the accurate analytical approximations are obtained for the first time, which are valid and convergent for all time t. This indicates the validity and great potential of the homotopy analysis method for solving nonlinear fractional integro–differential equations. 相似文献
10.
Dennis Kriventsov 《偏微分方程通讯》2013,38(12):2081-2106
We prove a C 1, α interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity of a special class of nonlinear operators. 相似文献
11.
Heinz‐Jürgen Flad Reinhold Schneider Bert‐Wolfgang Schulze 《Mathematical Methods in the Applied Sciences》2008,31(18):2172-2201
We study the asymptotic regularity of solutions to Hartree–Fock (HF) equations for Coulomb systems. To deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo‐differential operators on conical manifolds. First, the non‐self‐consistent‐field case is considered, which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator, which guarantee ellipticity for the local part of the Fock operator on the open stretched cone ?+ × S2. This proves the existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed‐point approach based on Cancès and Le Bris analysis of the level‐shifting algorithm, we show via another bootstrap argument that the corresponding self‐consistent‐field solutions to the HF equation have the same type of asymptotic regularity. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
12.
Jonatan Lenells 《Studies in Applied Mathematics》2016,136(1):3-63
The unified transform method introduced by Fokas can be used to analyze initial‐boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier transforms and the formulation of a Riemann‐Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier transforms. Using the theory of L2‐RH problems, we consider the construction of quarter plane solutions which are C1 in time and C3 in space. 相似文献
13.
Berna Bülbül Mustafa Gülsu Mehmet Sezer 《Numerical Methods for Partial Differential Equations》2010,26(5):1006-1020
The aim of this article is to present an efficient numerical procedure for solving nonlinear integro‐differential equations. Our method depends mainly on a Taylor expansion approach. This method transforms the integro‐differential equation and the given conditions into the matrix equation which corresponds to a system of nonlinear algebraic equations with unkown Taylor coefficients. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer program written in Maple10. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 相似文献
14.
K. Parand A. R. Rezaei A. Taghavi 《Mathematical Methods in the Applied Sciences》2010,33(17):2076-2086
This paper aims to compare rational Chebyshev (RC) and Hermite functions (HF) collocation approach to solve Volterra's model for population growth of a species within a closed system. This model is a nonlinear integro‐differential equation where the integral term represents the effect of toxin. This approach is based on orthogonal functions, which will be defined. The collocation method reduces the solution of this problem to the solution of a system of algebraic equations. We also compare these methods with some other numerical results and show that the present approach is applicable for solving nonlinear integro‐differential equations. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
15.
Richard Ewing Raytcho Lazarov Yanping Lin 《Numerical Methods for Partial Differential Equations》2000,16(3):285-311
In this article, we study finite volume element approximations for two‐dimensional parabolic integro‐differential equations, arising in the modeling of nonlocal reactive flows in porous media. These types of flows are also called NonFickian flows and exhibit mixing length growth. For simplicity, we consider only linear finite volume element methods, although higher‐order volume elements can be considered as well under this framework. It is proved that the finite volume element approximations derived are convergent with optimal order in H1‐ and L2‐norm and are superconvergent in a discrete H1‐norm. By examining the relationship between finite volume element and finite element approximations, we prove convergence in L∞‐ and W1,∞‐norms. These results are also new for finite volume element methods for elliptic and parabolic equations. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 285–311, 2000 相似文献
16.
Pointwise Estimates and Regularity in Geometric Optics and Other Generated Jacobian Equations
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The study of reflector surfaces in geometric optics necessitates the analysis of certain nonlinear equations of Monge‐Ampère type known as generated Jacobian equations. This class of equations, whose general existence theory has been recently developed by Trudinger, goes beyond the framework of optimal transport. We obtain pointwise estimates for weak solutions of such equations under minimal structural and regularity assumptions, covering situations analogous to those of costs satisfying the A3‐weak condition introduced by Ma, Trudinger, and Wang in optimal transport. These estimates are used to develop a C1,α regularity theory for weak solutions of Aleksandrov type. The results are new even for all known near‐field reflector/refractor models, including the point source and parallel beam reflectors, and are applicable to problems in other areas of geometry, such as the generalized Minkowski problem.© 2017 Wiley Periodicals, Inc. 相似文献
17.
《随机分析与应用》2013,31(5):1115-1139
Abstract We establish the global existence and uniqueness of mild solutions for a class of first‐order abstract stochastic Sobolev‐type integro‐differential equations in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time, t, but also on the corresponding probability distribution at time t. Results concerning the continuous dependence of solutions on the initial data and almost sure exponential stability, as well as an extension of the existence result to the case in which the classical initial condition is replaced by a so‐called nonlocal initial condition, are also discussed. Finally, an example is provided to illustrate the applicability of the general theory. 相似文献
18.
Pseudo asymptotically periodic mild solutions of semilinear functional integro‐differential equations in Banach spaces
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Zhinan Xia Dingjiang Wang Ching‐Feng Wen Jen‐Chih Yao 《Mathematical Methods in the Applied Sciences》2017,40(18):7333-7355
In this paper, we introduce and investigate the functions of (μ,ν)‐pseudo S‐asymptotically ω‐periodic of class r(class infinity). We systematically explore the properties of these functions in Banach space including composition theorems. As applications, we establish some sufficient criteria for (μ,ν)‐pseudo S‐asymptotic ω‐periodicity of (nonautonomous) semilinear integro‐differential equations with finite or infinite delay. Finally, some interesting examples are presented to illustrate the main findings. 相似文献
19.
We introduce the concept of Lp-maximal regularity for second order Cauchy problems. We prove Lp-maximal regularity for an abstract model problem and we apply the abstract results to prove existence, uniqueness and regularity
of solutions for nonlinear wave equations.
The author acknowledges with thanks the support provided by the Department ofApplied Analysis, University of Ulm, and the
travel grants provided by NBMH India and MSF Delhi, India. 相似文献
20.
It is shown that the geometrically correct investigation of regularity of nonlinear differential flows on manifolds and related
parabolic equations requires the introduction of a new type of variations with respect to the initial data. These variations
are defined via a certain generalization of a covariant Riemannian derivative to the case of diffeomorphisms. The appearance
of curvature in the structure of high-order variational equations is discussed and a family of a priori nonlinear estimates of regularity of any order is obtained. By using the relationship between the differential equations
on manifolds and semigroups, we study C
∞-regular properties of solutions of the parabolic Cauchy problems with coefficients increasing at infinity. The obtained conditions
of regularity generalize the classical coercivity and dissipation conditions to the case of a manifold and correlate (in a
unified way) the behavior of diffusion and drift coefficients with the geometric properties of the manifold without traditional
separation of curvature.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1011–1034, August, 2006. 相似文献