An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh

Das, P. C.

doi:10.1007/s11075-018-0557-4

Cited by 98 publications

(35 citation statements)

References 19 publications

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“…When μ the shear modulus is small, then this kind of problems is called singularly perturbed problems, where the uniform mesh or L2 norm-based error analysis does not converge to the original problem, and hence, one must use adaptive mesh with uniform error analysis. This numerical analysis is given in the papers [8][9][10][11][12][13][14].…”

Section: Governing Equationmentioning

confidence: 99%

WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition

Koubaiti¹,

Elkhalfi²,

El-Mekkaoui

2020

Abstract and Applied Analysis

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The objective of this article is to discuss the existence and the uniqueness of a weighted extended B-spline- (WEB-spline-) based discrete solution for the 2D Navier-Lamé equation of linear elasticity with a different type of mixed boundary condition called CA,B boundary condition. Along with the usual weak mixed formulation, we give existence and uniqueness results for weak solution. Then, we illustrate the performance of Ritz–Galerkin schemes for a model problem and applications in linear elasticity. Finally, we discuss several implementation aspects. The numerical tests confirm that, due to the new integration routines, the weighted B-spline solvers have become considerably more efficient.

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Section: Governing Equationmentioning

confidence: 99%

WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition

Koubaiti¹,

Elkhalfi²,

El-Mekkaoui

2020

Abstract and Applied Analysis

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“…We have provided sufficient conditions under which the present model problem has a unique solution. In addition, we have considered a semi-analytical approach that can be considered as an alternative to other approaches presented in several works, [9][10][11][12][13] based on the homotopy perturbation technique to approximate the solutions of the model equations.…”

Section: Resultsmentioning

confidence: 99%

“…Here, we are interested in obtaining the solution of fractional nonlinear integro-differential equations by using semi-analytical homotopy-based perturbation methods. [6][7][8] This approach makes the approximation simple for complicated models and can be considered as an alternative way to avoid the numerical discretizations provided in other works [9][10][11][12][13][14][15] for perturbation problems. Furthermore, the present method is not only an effective tool for approximation of the solution but it can also provide the exact solution of certain problems.…”

Section: Introductionmentioning

confidence: 99%

Homotopy perturbation method for solving Caputo‐type fractional‐order Volterra‐Fredholm integro‐differential equations

Das

Rana

Ramos

2019

Comp and Math Methods

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This work considers the solution of fractional Volterra‐Fredholm integro‐differential equations. Here, we consider the approximation of the solution based on semi‐analytical approaches. We use the homotopy perturbation method approach for this purpose. It is observed through different examples that the adopted strategy is not only an effective tool for approximation of the solution but also can lead to the exact solution of certain problems.

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“…Note: Algorithms have been produced lately to solve ODEs and PDEs by moving grids, the latest papers which show its effectiveness can be read in [19], [20], [21], [22], [23], [24], [32] and [33]. These references provide the technique that show the moving mesh algorithm and the discrete problem can be patched up to find the solution on adaptive mesh.…”

Section: Numerical Grid Generationmentioning

confidence: 99%

Optimal Control Approach to Image Registration

Salako¹

2018

ujam

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We discuss the orthogonality problem of moving grids, where we minimize a cost function with a regularization term and improve the orthogonality of grids by making the angle between grids close to 90  .The initial grid used is obtained by a well-established method known as the grid deformation method. We will then replace the cost function in the orthogonality problem with sum of squared differences (SSD) to discuss the image registration problem. We will discuss the non-uniqueness of solutions, existence of optimal solutions and prove the existence of Lagrange multipliers of the image registration problem using the Direct Method in Calculus of Variation and then derive an optimality system based on the construction of a Lagrangian functional from which optimal transformations can be calculated.

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An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh

Cited by 98 publications

References 19 publications

WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition

WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition

Homotopy perturbation method for solving Caputo‐type fractional‐order Volterra‐Fredholm integro‐differential equations

Optimal Control Approach to Image Registration

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