Numerical solution of singularly perturbed boundary value problems by improved least squares method

Ahmadinia, M.; Safari, Zeinab

doi:10.1016/j.cam.2017.09.023

Cited by 12 publications

(5 citation statements)

References 20 publications

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“…The maximum errors in (41) for this example are tabulated in table 3 for various values of N with fixed value = 10 −2 and compared with results in [1]. Also, table 4 displays E N for various values of , this table shows that by increasing N the errors in our scheme decrease.…”

Section: Numerical Resultsmentioning

confidence: 94%

“…To obtain a matrix representation of the equations (18), let I (i) , i = 0, 1, 2 be the m × m matrices whose jkth entry is given by ( 9)- (11). Note that the matrix I (2) and I (1) are symmetric and skew-symmetric matrices respectively, also I (0) is identity matrix. We define the m × m diagonal matrix as follow:…”

Section: The Sinc-collocation Methodsmentioning

confidence: 99%

“…In recent years, various numerical methods have been introduced and developed to solve the singularly perturbed differential equations such as least squares method [1], the finite difference scheme and Shishkin mesh [16], a homotopy technique [22], collocation method with Haar wavelets [21] and Bessel functions of the first kind of singularly perturbed differential equation [23]. A singularly perturbed problem having two parameters with discontinuous source term was considered in [7].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A numerical technique for solving a class of nonlinear singularly perturbed boundary value problems

Barati

Nabati

2021

Filomat

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In this work, we have studied a numerical scheme based on Sinc collocation method to solve a class of nonlinear singularly perturbed boundary value problems. The solution of the problems exhibit a boundary layer on the both sides or one side of the domain due to the presence of perturbation parameter ?. The Sinc method can control the oscillations in computed solutions at boundary layer regions naturally because the distribution of Sinc points is denser at near the boundaries. The convergence analysis is discussed and the method is shown to be an exponential convergent. The numerical results support the theoretical results and illustrate the efficiency and accuracy of the method compared with the results in the existing methods.

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Section: Numerical Resultsmentioning

confidence: 94%

Section: The Sinc-collocation Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A numerical technique for solving a class of nonlinear singularly perturbed boundary value problems

Barati

Nabati

2021

Filomat

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“…The PDEs with singular perturbation have been broadly studied by many scholars, including least squares method in [2], finite difference scheme in [4,26], Galerkin finite element method in [19], domain decomposition scheme in [20], reproducing kernel method(RKM) in [12] and others [6,27,30]. There are many references on numerical methods and numerical stability for delay differential equations, such as [5,15,17] to just list a few.…”

Section: Introductionmentioning

confidence: 99%

A Reproducing Kernel Method for Solving Singularly Perturbed Delay Parabolic Partial Differential Equations

Xie,

Zhang,

Niu

et al. 2023

Mathematical Modelling and Analysis

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In this article, we put forward an efficient method on the foundation of a few reproducing kernel spaces(RK-spaces) and the collocation method to seek the solution of delay parabolic partial differential equations(PDEs) with singular perturbation. The approximated solution to the equations is formulated and proved the exact solution is uniformly convergent by the solution. Furthermore, the partial differentiation of the approximated solution is also proved the partial derivatives of the exact solution is uniformly convergent by the solution. Meanwhile, we show that the accuracy of our method is in the order of T/n where T is the final time and n is the number of spatial (and time) discretization in the domain of interests. Three numerical examples are put forward to demonstrate the effectiveness of our presented scheme.

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“…Since the singular system model [1] is proposed by Rosenbrock in the early 1970s, the singular system has been widely studied, and many results are obtained [2,3]. Especially after the 1980s, singular systems, as an appropriate tool to deal with large-scale complex systems with multiobjective and multidimensional, have been applied to the fields of singularly perturbed theory [4], large-scale systems theory [5], circuit system [6], and so on.…”

Section: Introductionmentioning

confidence: 99%

Robust Preview Control for Uncertain Discrete Singular Systems

Liao

Shao

et al. 2018

Mathematical Problems in Engineering

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In this paper, the problem of robust preview control for uncertain discrete singular systems is considered. First of all, by employing the forward difference for uncertain discrete singular systems, the singular augmented error system with the state vector, the input control vector, and the previewable reference signal is derived. Since there is a singular matrix in the system, the existing method cannot be directly applied to this problem. By considering the stability of the transposition system with Linear Matrix Inequality (LMI) method, a new stability criterion for the transposition system is introduced. Then, the robust controller for the augmented error system is obtained, which is regarded as the robust preview controller for the original singular system. At last, the numerical simulation shows the correctness and effectiveness of the results.

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Numerical solution of singularly perturbed boundary value problems by improved least squares method

Cited by 12 publications

References 20 publications

A numerical technique for solving a class of nonlinear singularly perturbed boundary value problems

A numerical technique for solving a class of nonlinear singularly perturbed boundary value problems

A Reproducing Kernel Method for Solving Singularly Perturbed Delay Parabolic Partial Differential Equations

Robust Preview Control for Uncertain Discrete Singular Systems

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