RESEARCH PAPERS

Partial Differential Equations




Multiquadric radial basis function approximation methods for the numerical solution of partial differential equations
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Radial Basis Function (RBF) methods have become the primary tool for interpolating multidimensional scattered data. RBF methods also have become important tools for solving Partial Differential Equations (PDEs) in complexly shaped domains.

Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type
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ABSTRACT A class of quasilinear stochastic partial differential equations (SPDEs), driven by spatially correlated Brownian noise, is shown to become macroscopic (ie, deterministic), as the length of the correlations tends to 0. The limit is the solution of a quasilinear partial 

Homotopy perturbation method for solving partial differential equations
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We apply a relatively new technique which is called the homotopy perturbation method (HPM) for solving linear and nonlinear partial differential equations. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The 

Application of New Transform" Elzaki Transform" to Partial Differential Equations
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ABSTRACT The ELzaki transform of partial derivatives is derived, and its applicability demonstrated using four different partial differential equations. In this paper we find the particular solutions of these equations. Keywords: Elzaki Transform-Partial Differential 

Numerics for the Control of Partial Differential Equations
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ABSTRACT In this article we briefly present some aspects of the state of the art on efficient numerical approximation methods for control problems involving Partial Differential Equations. We focus mainly on the wave equation, as a paradigm of model for vibrations, 

The solution of the linear fractional partial differential equations using the homotopy analysis method
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In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with 

Partial Differential Equations MATH 334, Spring 2012; T-Th 9: 35–10: 50, Stevenson SC 1431
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The course is an interplay of Partial Differential Equations with an array of analytical techniques needed to their weak solvability. Solvability of linear and quasi-linear elliptic equations with measurable coefficients, motivates the theory of Sobolev spaces, traces 

New Application of Variational Iteration Method for Analytic Treatment of Nonlinear Partial Differential Equations
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ABSTRACT In this paper, numerical solutions of nonlinear Klein-Gordon equations and some other nonlinear partial differential equations are obtained by a new application of He's variational iteration method. An efficient way of choosing the initial approximation is 

A review of numerical methods for nonlinear partial differential equations
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ABSTRACT Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid-1940s. In a 1949 letter von Neumann wrote``the entire computing machine is merely one component of a greater 

Operator splitting for partial differential equations with Burgers nonlinearity
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ABSTRACT We provide a new analytical approach to operator splitting for equations of the type ut= Au+ uux where A is a linear differential operator such that the equation is well-posed. Particular examples include the viscous Burgers equation, the Korteweg–de Vries (KdV) 

Conditional distributions, exchangeable particle systems, and stochastic partial differential equations
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ABSTRACT Stochastic partial differential equations (SPDEs) whose solutions are probabilitymeasure-valued processes are considered. Measure-valued processes of this type arise naturally as de Finetti measures of infinite exchangeable systems of particles 

Entropy and partial differential equations
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A good many times I have been present at gatherings of people who, by the standards of traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been 

Finite element approximation of elliptic partial differential equations on implicit surfaces
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ABSTRACT The aim of this paper is to investigate finite element methods for the solution of elliptic partial differential equations on implicitly defined surfaces. The problem of solving such equations without triangulating surfaces is of increasing importance in various 

On the Coupling of Homotopy Perturbation and Laplace Transformation for System of Partial Differential Equations
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ABSTRACT The aim of this article is to propose a new algorithm, namely homotopy perturbation transform algorithm (HPTA). This new algorithm provides us with a convenient way to find exact solution with less computation as compared with standard homotopy perturbation 

A fast direct solver for a class of elliptic partial differential equations
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ABSTRACT We describe a fast and robust method for solving the large sparse linear systems that arise upon the discretization of elliptic partial differential equations such as Laplace's equation and the Helmholtz equation at low frequencies. While most existing fast schemes 

Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations
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ABSTRACT Forward, backward and elliptic Harnack inequalities for non–negative solutions of a class of singular, quasilinear, parabolic equations, are established. These classes of singular equations include the p–Laplacean equation and equations of the porous 

The solution of the variable coefficients fourth-order parabolic partial differential equations by the homotopy perturbation method
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In this work, the homotopy perturbation method proposed by Ji-Huan He is applied to solve both linear and nonlinear boundary value problems for fourth-order partial differential equations. The numerical results obtained with minimum amount of computation are 

Solving Partial Differential Equations and variational problems with networks of spiking neurons
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Many problems in computational and scientific disciplines can be represented as optimization problems and/or by partial differential equations. Partial differential equations contain both spatial and temporal derivatives and arise in physics, chemistry, fluid 

Certified reduced basis approximation for parametrized partial differential equations and applications
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ABSTRACT Reduction strategies, such as model order reduction (MOR) or reduced basis (RB) methods, in scientific computing may become crucial in applications of increasing complexity. In this paper we review the reduced basis method (built upon a high-fidelity"  

Adaptive wavelet methods for elliptic stochastic partial differential equations
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ABSTRACT We study the Besov regularity as well as linear and nonlinear approximation of random functions on bounded Lipschitz domains in Rd. The random functions are given either (i) explicitly in terms of a wavelet expansion or (ii) as the solution of a Poisson 

On elliptic partial differential equations with random coefficients
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ABSTRACT We consider stationary diffusion equations with random coefficients which cannot be bounded strictly away from zero and infinity by constants. We prove the existence of a unique solution to the corresponding weak formulation with different solution and test 

Wavelet methods for elliptic partial differential equations
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Wavelets have become a powerful tool in signal and image processing over more then a decade by now. Their use for the numerical solution of operator equations has been investigated more recently. By now the theoretical understanding of such methods is quite 

An Efficient Numerical Method for Solving Linear and Nonlinear Partial Differential Equationsby Combining Homotopy Analysis and Transform Method
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ABSTRACT In this article, we propose a reliable combination of Homotopy analysis method (HAM) and Laplace decomposition method (LDM) to solve linear and nonlinear partial differential equations effectively with less computation. The proposed method is called 

Recent developments in spectral stochastic methods for the numerical solution of stochasticpartial differential equations
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ABSTRACT Uncertainty quantification appears today as a crucial point in numerous branches of science and engineering. In the last two decades, a growing interest has been devoted to a new family of methods, called spectral stochastic methods, for the propagation of 

Isogeometric analysis for second order partial differential equations on surfaces
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ABSTRACT We consider the numerical solution of second order Partial Differential Equations (PDEs) on lower dimensional manifolds, specifically on surfaces in three dimensional spaces. For the spatial approximation, we consider Isogeometric Analysis which facilitates 

Converting fractional differential equations into partial differential equations
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A transform is suggested in this paper to convert fractional differential equations with the modified Riemann-Liouville derivative into partial differential equations, and it is concluded that the fractional order in fractional differential equations is equivalent to the fractal 

White noise analysis for stochastic partial differential equations
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ABSTRACT Stochastic partial differential equations arise when modelling uncertain phenomena. Here the emphasis is on uncertain systems where the randomness is spatial. In contrast to traditional slow computational approaches like Monte Carlo simulation, the 

Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction?
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ABSTRACT A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system 

On Besov regularity of solutions to nonlinear elliptic partial differential equations
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ABSTRACT In this paper, we study the regularity of the solutions to nonlinear elliptic equations. In particular, we are interested in smoothness estimates in the specific scale, t=(a/d+ 1/2)- 1, of Besov spaces which determines the approximation order of adaptive and other 

Introduction to the Theory of Stochastic Differential Equations and Stochastic Partial Differential Equations
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After briefly summarizing the basic concepts and facts in probability theory such as probability spaces, random variables, their convergences, independence, the Central Limit Theorem and Gaussian distributions, the Brownian motion will be introduced. The 

Hhere Analysis II: Partial Differential Equations
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course: Thursday: 9:15–10:45 h, Room 1.013 (Lecture hall 100, RUD 25) 13:15–14:45 h, 

Application of new transform" tarig transform" to partial differential equations
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ABSTRACT In this paper we derive the formulate for Tarig transform of partial derivatives and apply them to solve five types of initial value problems. Our purpose here is to show that the applicability of this interesting new transform and its effecting to solve such problems.

Introduction to Classical Partial Differential Equations
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The symbols N, R, C denote the natural numbers (including zero), all real numbers and all complex numbers, respectively. The symbols N, R, C denote the corresponding sets from which 0 has been excluded. We call x 2 R positive (negative) if x 0 (x 6 0/. In addition, we 


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