Solving nonlocal initial-boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations by the Adomian decomposition method

Bougoffa, Lazhar; Rach, Randolph

doi:10.1016/j.amc.2013.09.011

Cited by 16 publications

(7 citation statements)

References 32 publications

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“…Remark 7. It seems challenging to collaborate and compare numerically the presented method with other novel methods developed to solve parabolic equations [10,11,12]. The hypothesis is that a) our method will be faster on small times of evolution t because it does not involve numerical integration or matrix inversion b) our method needs to take more approximation steps (greater n) for large t as t/n appears in the final formula (4).…”

Section: Technique Employedmentioning

confidence: 99%

Approximations to the solution of Cauchy problem for a linear evolution equation via the space shift operator (second-order equation example)

Remizov¹

2018

Applied Mathematics and Computation

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We present a general method of solving the Cauchy problem for a linear parabolic partial differential equation of evolution type with variable coefficients and demonstrate it on the equation with derivatives of orders two, one and zero. The method is based on the Chernoff approximation procedure applied to a specially constructed shift operator. It is proven that approximations converge uniformly to the exact solution. Keywords:Cauchy problem, linear parabolic PDE, approximate solution, shift operator, Chernoff theorem, numerical method 2000 MSC: 35A35, 35C99, 35K15, 35K30 Problem setting and approach proposedConsider x ∈ R 1 , t ≥ 0 and set the Cauchy problem for a second-order parabolic partial differential equationThe coefficients a, b, c, u 0 above are bounded, uniformly continuous functions R 1 → R 1 . This paper is dedicated to deriving of an explicit formula

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Section: Technique Employedmentioning

confidence: 99%

Approximations to the solution of Cauchy problem for a linear evolution equation via the space shift operator (second-order equation example)

Remizov¹

2018

Applied Mathematics and Computation

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“…Following this pattern as well as evaluating two equations for every mid-point leads to (2 ) equations that can be placed in the first (2 ) rows of the matrix. The last two rows can be used to impose the boundary conditions given in (12) and (13). Following this manner, it is possible to group the above equations at every time interval +1 in the form of…”

Section: Analysis Of the Methodsmentioning

confidence: 99%

“…Traditional finite difference methods have been shown to have problems in accuracy [9]; therefore, a number of investigators have developed various numerical methods for the above problem. Recent results include methods based on -based finite difference [10], reproducing kernel space [11], and Adomian expansion [12].…”

Section: Introductionmentioning

confidence: 99%

A Numerical Method for 1-D Parabolic Equation with Nonlocal Boundary Conditions

Tadi

Radenković

2014

International Journal of Computational Mathematics

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This paper is concerned with a local method for the solution of one-dimensional parabolic equation with nonlocal boundary conditions. The method uses a coordinate transformation. After the coordinate transformation, it is then possible to obtain exact solutions for the resulting equations in terms of the local variables. These exact solutions are in terms of constants of integration that are unknown. By imposing the given boundary conditions and smoothness requirements for the solution, it is possible to furnish a set of linearly independent conditions that can be used to solve for the constants of integration. A number of examples are used to study the applicability of the method. In particular, three nonlinear problems are used to show the novelty of the method.

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“…Innumerable number of methods for obtaining analytical and approximate solutions to nonlinear evolution equations have been proposed. Some of the analytical methods that have been used to solve evolution nonlinear partial differential equations include Adomian's decomposition method [ 1 – 3 ], homotopy analysis method [ 4 – 7 ], tanh-function method [ 8 – 10 ], Haar wavelet method [ 11 – 13 ], and Exp-function method [ 14 – 16 ]. Several numerical methods have been used to solve nonlinear evolution partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

A Bivariate Chebyshev Spectral Collocation Quasilinearization Method for Nonlinear Evolution Parabolic Equations

Motsa

Magagula

Sibanda

2014

The Scientific World Journal

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This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.

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Solving nonlocal initial-boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations by the Adomian decomposition method

Cited by 16 publications

References 32 publications

Approximations to the solution of Cauchy problem for a linear evolution equation via the space shift operator (second-order equation example)

Approximations to the solution of Cauchy problem for a linear evolution equation via the space shift operator (second-order equation example)

A Numerical Method for 1-D Parabolic Equation with Nonlocal Boundary Conditions

A Bivariate Chebyshev Spectral Collocation Quasilinearization Method for Nonlinear Evolution Parabolic Equations

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