共查询到20条相似文献,搜索用时 46 毫秒
1.
《Journal of Computational and Applied Mathematics》2006,197(2):406-420
In this paper we study the numerical solution of parabolic Volterra integro-differential equations on certain unbounded two-dimensional spatial domains. The method is based on the introduction of a feasible artificial boundary and the derivation of corresponding artificial (fully transparent) boundary conditions. Two examples illustrate the application and numerical performance of the method. 相似文献
2.
S. Brzychczy 《Mathematical and Computer Modelling》2002,36(11-13)
We consider the Fourier first initial-boundary value problem for a weakly coupled infinite system of semilinear parabolic differential-functional equations of reaction-diffusion type in arbitrary (bounded or unbounded) domain. The right-hand sides of the system are functionals of unknown functions of the Volterra type. Differential-integral equations give examples of such equations. To prove the existence and uniqueness of the solutions, we apply the monotone iterative method. The underlying monotone iterative scheme can be used for the computation of numerical solution. 相似文献
3.
We report a new parallel iterative algorithm for semi-linear parabolic partial differential equations (PDEs) by combining a kind of waveform relaxation (WR) techniques into the classical parareal algorithm. The parallelism can be simultaneously exploited by WR and parareal in different directions. We provide sharp error estimations for the new algorithm on bounded time domain and on unbounded time domain, respectively. The iterations of the parareal and the WR are balanced to optimize the performance of the algorithm. Furthermore, the speedup and the parallel efficiency of the new approach are analyzed. Numerical experiments are carried out to verify the effectiveness of the theoretic work. 相似文献
4.
5.
The aim of this paper is to present an efficient numerical procedure for solving the two-dimensional nonlinear Volterra integro-differential
equations (2-DNVIDE) by two-dimensional differential transform method (2-DDTM). The technique that we used is the differential
transform method, which is based on Taylor series expansion. Using the differential transform, 2-DNVIDE can be transformed
to algebraic equations, and the resulting algebraic equations are called iterative equations. New theorems for the transformation
of integrals and partial differential equations are introduced and proved. The reliability and efficiency of the proposed
scheme are demonstrated by some numerical experiments. 相似文献
6.
We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method. 相似文献
7.
Fractional calculus is an extension of derivatives and integrals to non-integer orders and has been widely used to model scientific and engineering problems. In this paper, we describe the fractional derivative in the Caputo sense and give the second kind Chebyshev wavelet (SCW) operational matrix of fractional integration. Then based on above results we propose the SCW operational matrix method to solve a kind of nonlinear fractional-order Volterra integro-differential equations. The main characteristic of this approach is that it reduces the integro-differential equations into a nonlinear system of algebraic equations. Thus, it can simplify the problem of fractional order equation solving. The obtained numerical results indicate that the proposed method is efficient and accurate for this kind equations. 相似文献
8.
Mostafa Mahmoudi Mehdi Ghovatmand Mohammad Hadi Noori Skandari 《Mathematical Methods in the Applied Sciences》2020,43(5):2357-2368
In this paper, we suggest a convergent numerical method for solving nonlinear delay Volterra integro-differential equations. First, we convert the problem into a continuous-time optimization problem and then use a shifted pseudospectral method to discrete the problem. Having solved the last problem, we can achieve the pointwise and continuous approximate solutions for the main delay Volterra integro-differential equations. Here, we analyze the convergence of the method and solve some numerical examples to show the efficiency of the method. 相似文献
9.
In this paper, a new method for solving arbitrary order ordinary differential equations and integro-differential equations of Fredholm and Volterra kind is presented. In the proposed method, these equations with separated boundary conditions are converted to a parametric optimization problem subject to algebraic constraints. Finally, control and state variables will be approximated by a Chebychev series. In this method, a new idea has been used, which offers us the ability of applying the mentioned method for almost all kinds of ordinary differential and integro-differential equations with different types of boundary conditions. The accuracy and efficiency of the proposed numerical technique have been illustrated by solving some test problems. 相似文献
10.
In this paper, we are concerned with splitting methods for the time integration of abstract evolution equations. We introduce
an analytic framework which allows us to prove optimal convergence orders for various splitting methods, including the Lie
and Peaceman–Rachford splittings. Our setting is applicable for a wide variety of linear equations and their dimension splittings.
In particular, we analyze parabolic problems with Dirichlet boundary conditions, as well as degenerate equations on bounded
domains. We further illustrate our theoretical results with a set of numerical experiments.
This work was supported by the Austrian Science Fund under grant M961-N13. 相似文献
11.
Summary. In this paper we consider the numerical simulations of the incompressible materials on an unbounded domain in . A series of artificial boundary conditions at a circular artificial boundary for solving incompressible materials on an
unbounded domain is given. Then the original problem is reduced to a problem on a bounded domain, which be solved numerically
by a mixed finite element method. The numerical example shows that our artificial boundary conditions are very effective.
ReceivedJune 7, 1995 / Revised version received August 19, 1996 相似文献
12.
Summary. The boundary element method (BEM) is of advantage in many applications including far-field computations in magnetostatics
and solid mechanics as well as accurate computations of singularities. Since the numerical approximation is essentially reduced
to the boundary of the domain under consideration, the mesh generation and handling is simpler than, for example, in a finite
element discretization of the domain. In this paper, we discuss fast solution techniques for the linear systems of equations
obtained by the BEM (BE-equations) utilizing the non-overlapping domain decomposition (DD). We study parallel algorithms for
solving large scale Galerkin BE–equations approximating linear potential problems in plane, bounded domains with piecewise
homogeneous material properties. We give an elementary spectral equivalence analysis of the BEM Schur complement that provides
the tool for constructing and analysing appropriate preconditioners. Finally, we present numerical results obtained on a massively
parallel machine using up to 128 processors, and we sketch further applications to elasticity problems and to the coupling
of the finite element method (FEM) with the boundary element method. As shown theoretically and confirmed by the numerical
experiments, the methods are of algebraic complexity and of high parallel efficiency, where denotes the usual discretization parameter.
Received August 28, 1996 / Revised version received March 10, 1997 相似文献
13.
Time discretization via Laplace transformation of an integro-differential equation of parabolic type
We consider the discretization in time of an inhomogeneous parabolic integro-differential equation, with a memory term of
convolution type, in a Banach space setting. The method is based on representing the solution as an integral along a smooth
curve in the complex plane which is evaluated to high accuracy by quadrature, using the approach in recent work of López-Fernández
and Palencia. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved
in parallel. The method is combined with finite element discretization in the spatial variables to yield a fully discrete
method. The paper is a further development of earlier work by the authors, which on the one hand treated purely parabolic
equations and, on the other, an evolution equation with a positive type memory term.
The authors acknowledge the support of the Australian Research Council. 相似文献
14.
A.S. Berdyshev 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(6):3268-3273
In the present work, a non-local boundary value problem with special gluing conditions for a mixed parabolic-hyperbolic equation with parameter is considered. The parabolic part of this equation is a fractional analogue of heat equation and the hyperbolic part is the telegraph equation. The considered problem is reduced, for positive values of the parameter, to an equivalent system of the second kind Volterra integral equations. Due to the influence of the fractional diffusion equation, the looked for solution belongs to a specific class of functions. The method of the Green functions and the properties of integro-differential operators are on the basis of the investigation. 相似文献
15.
《Communications in Nonlinear Science & Numerical Simulation》2011,16(3):1216-1226
A computational method for numerical solution of a nonlinear Volterra integro-differential equation of fractional (arbitrary) order which is based on CAS wavelets and BPFs is introduced. The CAS wavelet operational matrix of fractional integration is derived and used to transform the main equation to a system of algebraic equations. Some examples are included to demonstrate the validity and applicability of the technique. 相似文献
16.
Xiao-Chuan Cai 《Numerische Mathematik》1991,60(1):41-61
Summary In this paper, we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems. We introduce three additive Schwarz type domain decomposition methods for general, not necessarily selfadjoint, linear, second order, parabolic partial differential equations and also study the convergence rates of these algorithms. The resulting preconditioned linear system of equations is solved by the generalized minimal residual method. Numerical results are also reported.This work was supported in part by the National Science Foundation under Grant NSF-CCR-8903003 at the Courant Institute, New York University and in part by the National Science Foundation under contract number DCR-8521451 and ECS-8957475 at Yale University 相似文献
17.
The main purpose of this work is to provide a numerical method for the solution of Volterra functional integro-differential equations of neutral type based on a spectral approach. We analyze the convergence properties of the spectral method to approximate smooth solutions of Volterra functional integro-differential equations of neutral type. It is shown that for the neutral integro-differential equations, the spectral methods yield an exponential order of convergence. 相似文献
18.
Lalit Kumar Sivaji Ganesh Sista Konijeti Sreenadh 《Mathematical Methods in the Applied Sciences》2020,43(15):9129-9150
The aim of this paper is to study parabolic integro-differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results. 相似文献
19.
In this paper, the problem of solving the parabolic partial differential equations subject to given initial and nonlocal boundary conditions is considered. We change the problem to a system of Volterra integral equations of convolution type. By using Sinc-collocation method, the resulting integral equations are replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the condition number of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. Some examples are considered to illustrate the ability of this method. 相似文献
20.
In this paper we consider a hyperbolic equation, with a memory term in time, which can be seen as a singular perturbation of the heat equation with memory. The qualitative properties of the solutions of the initial boundary value problems associated with both equations are studied. We propose numerical methods for the hyperbolic and parabolic models and their stability properties are analyzed. Finally, we include numerical experiments illustrating the performance of those methods. 相似文献