An operational Haar wavelet collocation method for solving singularly perturbed boundary-value problems

Shah, Firdous A.; Abass, R.

doi:10.1007/s40324-016-0094-9

Cited by 10 publications

(5 citation statements)

References 27 publications

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“…15 A lot of researchers have adopted various Haar function-based approaches to solve different applied problems in science and engineering field. [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] The Haar wavelets have moreover been utilized to fathom integral equations [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48] and direct as well as inverse problems. [48][49][50][51] The Haar function has also been used in modern technology for detecting the software piracy.…”

Section: Introductionmentioning

confidence: 99%

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

Ahsan

Haq

et al. 2022

Math Methods in App Sciences

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In this discussion, a new numerical algorithm focused on the Haar wavelet is used to solve linear and nonlinear inverse problems with unknown heat source. The heat source is dependent on time and space variables. These types of inverse problems are ill‐posed and are challenging to solve accurately. The linearization technique converted the nonlinear problem into simple nonhomogeneous partial differential equation. In this Haar wavelet collocation method (HWCM), the time part is discretized by using finite difference approximation, and space variables are handled by Haar series approximation. The main contribution of the proposed method is transforming this ill‐posed problem into well‐conditioned algebraic equation with the help of Haar functions, and hence, there is no need to implement any sort of regularization technique. The results of numerical method are efficient and stable for this ill‐posed problems containing different noisy levels. We have utilized the proposed method on several numerical examples and have valuable efficiency and accuracy.

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Section: Introductionmentioning

confidence: 99%

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

Ahsan

Haq

et al. 2022

Math Methods in App Sciences

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“…Different algorithms, based on weak and strong formulations, include meshless wavelet method, 28 Daubechies wavelet‐based method, 29 wavelet Galerkin method, 30 wavelet collocation methods, 31,32 and Chebyshev wavelet method 33 . A thorough introduction of wavelet schemes for PDEs is given in Dahmen et al 34 Different forms of ODEs, integro‐differential equations and PDEs that model different scientific and engineering phenomena, have been solved by Haar wavelets 35–52 . A further development of Haar wavelets is related with solution of challenging fractional differential and integral equations 53–57 .…”

Section: Introductionmentioning

confidence: 99%

A multiresolution collocation method and its convergence for Burgers' type equations

Ahsan

Tran

Islam

et al. 2022

Math Methods in App Sciences

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In this article, a hybrid numerical method based on Haar wavelets and finite differences is proposed for shock ridden evolutionary nonlinear time-dependent partial differential equations (PDEs). A linear procedure using Taylor expansions is adopted to linearize the nonlinearity. The Euler difference scheme is used to discretize the time derivative part of the PDE. The PDE is converted into full algebraic form, once the space derivatives are replaced by finite Haar series. Convergence analysis is performed both in space and time, and computational stability of the proposed method is also discussed. Different benchmark cases related to 1-D and 2-D Burgers' type equations are considered to verify the theory. The proposed method is also compared numerically with existing methods in the literature.

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“…The different algorithms, based on weak and strong formulations, are the meshless wavelet method [16], the Daubechies wavelet-based method [17], the wavelet Galerkin method [18], and the wavelet collocation methods [19,20].A thorough introduction of the wavelets schemes for partial differential equations (PDEs) is given in [21]. Different scientific and engineering phenomena have been represented in the forms of ordinary differential equations (ODEs), integro-differential equations, and PDEs, which have been solved by Haar wavelets in the references [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. A further development of the Haar wavelet is related to the solution of challenging fractional differential and integral equations [38][39][40][41][42].…”

Section: Introductionmentioning

confidence: 99%

Applications of Haar Wavelet-Finite Difference Hybrid Method and Its Convergence for Hyperbolic Nonlinear Schrödinger Equation with Energy and Mass Conversion

et al. 2021

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This article is concerned with the numerical solution of nonlinear hyperbolic Schro¨dinger equations (NHSEs) via an efficient Haar wavelet collocation method (HWCM). The time derivative is approximated in the governing equations by the central difference scheme, while the space derivatives are replaced by finite Haar series, which transform it to full algebraic form. The experimental rate of convergence follows the theoretical statements of convergence and the conservation laws of energy and mass are also presented, which strengthens the proposed method to be convergent and conservative. The Haar wavelets based on numerical results for solitary wave shape of |φ| are discussed in detail. The proposed approach provides a fast convergent approximation to the NHSEs. The reliability and efficiency of the method are illustrated by computing the maximum error norm and the experimental rate of convergence for different problems. Comparisons are performed with various existing methods in recent literature and better performance of the proposed method is shown in various tables and figures.

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An operational Haar wavelet collocation method for solving singularly perturbed boundary-value problems

Cited by 10 publications

References 27 publications

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

A multiresolution collocation method and its convergence for Burgers' type equations

Applications of Haar Wavelet-Finite Difference Hybrid Method and Its Convergence for Hyperbolic Nonlinear Schrödinger Equation with Energy and Mass Conversion

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