The Experts below are selected from a list of 312 Experts worldwide ranked by ideXlab platform
Kapil K. Sharma - One of the best experts on this subject based on the ideXlab platform.
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Uniformly convergent non-standard finite difference methods for singularly perturbed Differential-Difference Equations with delay and advance
International Journal for Numerical Methods in Engineering, 2006Co-Authors: Kailash C. Patidar, Kapil K. SharmaAbstract:A new class of fitted operator finite difference methods are constructed via non-standard finite difference methods ((NSFDM)s) for the numerical solution of singularly perturbed differential difference Equations having both delay and advance arguments. The main idea behind the construction of our method(s) is to replace the denominator function of the classical second-order derivative with a positive function derived systematically in such a way that it captures significant properties of the governing differential equation and thus provides the reliable numerical results. Unlike other FOFDMs constructed in standard ways, the methods that we present in this paper are fairly simple to construct (and thus enrich the class of fitted operator methods by adding these new methods). These methods are shown to be e-uniformly convergent with order two which is the highest possible order of convergence obtained via any fitted operator method for the problems under consideration. This paper further clarifies several doubts, e.g. why a particular scheme is not suitable for the whole range of values of the associated parameters and what could be the possible remedies. Finally, we provide some numerical examples which illustrate the theoretical findings. Copyright © 2005 John Wiley & Sons, Ltd.
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e uniformly convergent non standard finite difference methods for singularly perturbed differential difference Equations with small delay
Applied Mathematics and Computation, 2006Co-Authors: Kailash C. Patidar, Kapil K. SharmaAbstract:Abstract Non-standard finite difference methods (NSFDMs), now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineerings for which the existing methodologies do not give reliable results, these NSFDMs are solving them competitively. To this end, in this paper we consider, second order, linear, singularly perturbed differential difference Equations. Using the second of the five non-standard modeling rules of Mickens [R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994], the new finite difference methods are obtained for the particular cases of these problems. This rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation(s). Both theoretically and numerically, we show that these NSFDMs are e-uniformly convergent.
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ε-Uniformly convergent non-standard finite difference methods for singularly perturbed differential difference Equations with small delay
Applied Mathematics and Computation, 2006Co-Authors: Kailash C. Patidar, Kapil K. SharmaAbstract:Abstract Non-standard finite difference methods (NSFDMs), now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineerings for which the existing methodologies do not give reliable results, these NSFDMs are solving them competitively. To this end, in this paper we consider, second order, linear, singularly perturbed differential difference Equations. Using the second of the five non-standard modeling rules of Mickens [R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994], the new finite difference methods are obtained for the particular cases of these problems. This rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation(s). Both theoretically and numerically, we show that these NSFDMs are e-uniformly convergent.
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An ε-uniform convergent method for a general boundary-value problem for singularly perturbed Differential-Difference Equations: Small shifts of mixed type with layer behavior
Journal of Computational Methods in Sciences and Engineering archive, 2006Co-Authors: Mohan K. Kadalbajoo, Kapil K. SharmaAbstract:In this paper, we continue the study of singularly perturbed Differential-Difference Equations with small shifts, which is motivated by the problem of determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in the dendrites [1]. We consider a more general boundary-value problem which contains both convection and reaction terms with both type of shifts (negative as well as positive) than the problem discussed in paper [2,7]. We consider the case when the solution of such type of boundary-value problem exhibits boundary layer behavior. An e-uniform convergent scheme based on fitting operator is derived for boundary value problems for singularly perturbed Differential-Difference Equations with small shifts. We introduce an exponential fitting parameter to the standard finite difference scheme which reflects the singularly perturbed nature of differential operator. The method is analyzed for convergence. Several numerical experiments are carried in support of theoretical results and to show the effect of small shifts on the boundary layer solution.
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Numerical treatment for singularly perturbed nonlinear differential difference Equations with negative shift
Nonlinear Analysis-theory Methods & Applications, 2005Co-Authors: Mohan K. Kadalbajoo, Kapil K. SharmaAbstract:Abstract This paper is devoted to the numerical study of the boundary value problems for singularly perturbed nonlinear differential difference Equations with negative shift. The highest derivative of such types of boundary value problems is multiplied by a small parameter which is called the singular perturbation parameter. Due to the presence of the singular perturbation parameter the solution of such problems exhibits layer behavior and the classical numerical methods to approximate the solution do not converge uniformly with respect to the singular perturbation parameter. Thus, in general, to tackle the boundary value problems for singularly perturbed nonlinear differential difference Equations with negative shift, one encounters three difficulties: (i) due to the presence of nonlinearity, (ii) due to the presence of terms containing shifts and (iii) due to the presence of singular perturbation parameter. To resolve the first difficulty, we use quasilinearization process to linearize the nonlinear differential equation. After applying the quasilinearization process to the nonlinear problem, a sequence of linearized problems is obtained. We show that the solution of the sequence of the linearized problems converge quadratically to the solution of the original nonlinear problem. To resolve the second difficulty, Taylor's series is used to tackle the terms containing shift provided the shifts is of small order of singular perturbation parameter and when shift is of capital order of singular perturbation, a special type of mesh is used. Finally, to resolve the third difficulty, we use a piecewise uniform mesh which is dense in the boundary layer region and coarse in the outer region and standard finite difference operators are used to approximate the derivatives. The difference scheme so obtained is shown to be parameter uniform by establishing the parameter-uniform error estimates. To demonstrate the efficiency of the method, some numerical experiments are carried out.
Kailash C. Patidar - One of the best experts on this subject based on the ideXlab platform.
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Uniformly convergent non-standard finite difference methods for singularly perturbed Differential-Difference Equations with delay and advance
International Journal for Numerical Methods in Engineering, 2006Co-Authors: Kailash C. Patidar, Kapil K. SharmaAbstract:A new class of fitted operator finite difference methods are constructed via non-standard finite difference methods ((NSFDM)s) for the numerical solution of singularly perturbed differential difference Equations having both delay and advance arguments. The main idea behind the construction of our method(s) is to replace the denominator function of the classical second-order derivative with a positive function derived systematically in such a way that it captures significant properties of the governing differential equation and thus provides the reliable numerical results. Unlike other FOFDMs constructed in standard ways, the methods that we present in this paper are fairly simple to construct (and thus enrich the class of fitted operator methods by adding these new methods). These methods are shown to be e-uniformly convergent with order two which is the highest possible order of convergence obtained via any fitted operator method for the problems under consideration. This paper further clarifies several doubts, e.g. why a particular scheme is not suitable for the whole range of values of the associated parameters and what could be the possible remedies. Finally, we provide some numerical examples which illustrate the theoretical findings. Copyright © 2005 John Wiley & Sons, Ltd.
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e uniformly convergent non standard finite difference methods for singularly perturbed differential difference Equations with small delay
Applied Mathematics and Computation, 2006Co-Authors: Kailash C. Patidar, Kapil K. SharmaAbstract:Abstract Non-standard finite difference methods (NSFDMs), now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineerings for which the existing methodologies do not give reliable results, these NSFDMs are solving them competitively. To this end, in this paper we consider, second order, linear, singularly perturbed differential difference Equations. Using the second of the five non-standard modeling rules of Mickens [R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994], the new finite difference methods are obtained for the particular cases of these problems. This rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation(s). Both theoretically and numerically, we show that these NSFDMs are e-uniformly convergent.
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ε-Uniformly convergent non-standard finite difference methods for singularly perturbed differential difference Equations with small delay
Applied Mathematics and Computation, 2006Co-Authors: Kailash C. Patidar, Kapil K. SharmaAbstract:Abstract Non-standard finite difference methods (NSFDMs), now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineerings for which the existing methodologies do not give reliable results, these NSFDMs are solving them competitively. To this end, in this paper we consider, second order, linear, singularly perturbed differential difference Equations. Using the second of the five non-standard modeling rules of Mickens [R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994], the new finite difference methods are obtained for the particular cases of these problems. This rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation(s). Both theoretically and numerically, we show that these NSFDMs are e-uniformly convergent.
Khaled A. Gepreel - One of the best experts on this subject based on the ideXlab platform.
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Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics
Abstract and Applied Analysis, 2013Co-Authors: Khaled A. Gepreel, Taher A. Nofal, Fawziah M. AlotaibiAbstract:We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference Equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference Equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential Equations.
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Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics
Journal of Applied Mathematics, 2012Co-Authors: Khaled A. Gepreel, A. R. ShehataAbstract:We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference Equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference Equations in mathematical physics via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference Equations.
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The Modified Rational Jacobi Elliptic Functions Method for Nonlinear Differential Difference Equations
Journal of Applied Mathematics, 2012Co-Authors: Khaled A. Gepreel, Taher A. Nofal, Ali A. Al-thobaitiAbstract:We modified the rational Jacobi elliptic functions method to construct some new exact solutions for nonlinear differential difference Equations in mathematical physics via the lattice equation, the discrete nonlinear Schrodinger equation with a saturable nonlinearity, the discrete nonlinear Klein-Gordon equation, and the quintic discrete nonlinear Schrodinger equation. Some new types of the Jacobi elliptic solutions are obtained for some nonlinear differential difference Equations in mathematical physics. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference Equations.
Fawziah M. Alotaibi - One of the best experts on this subject based on the ideXlab platform.
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Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics
Abstract and Applied Analysis, 2013Co-Authors: Khaled A. Gepreel, Taher A. Nofal, Fawziah M. AlotaibiAbstract:We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference Equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference Equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential Equations.
I. A. Kolesnikova - One of the best experts on this subject based on the ideXlab platform.
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Necessary and sufficient conditions for the existence of symmetry groups for systems of Differential-Difference Equations with partial derivatives
Journal of Mathematical Sciences, 2012Co-Authors: I. A. KolesnikovaAbstract:In the present paper, we obtain necessary and sufficient conditions for the existence of symmetry groups for systems of Differential-Difference Equations with partial derivatives. Constructions of symmetry groups and recursion operators for various Differential-Difference Equations are described. The corresponding symmetry generators that transform solutions of given Equations into other solutions are obtained.
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On potentiality conditions for partial Differential-Difference Equations
Differential Equations, 2010Co-Authors: I. A. KolesnikovaAbstract:We obtain necessary and sufficient conditions for the existence of variational principles with respect to a classical bilinear form for boundary value problems for partial Differential-Difference Equations.
