A Haar wavelet multi-resolution collocation method for singularly perturbed differential equations with integral boundary conditions

Ahsan, Muhammad; Böhner, Martin; Ullah, Aizaz; Khan, Amir; Ahmad, Sheraz

doi:10.1016/j.matcom.2022.08.004

Cited by 20 publications

(4 citation statements)

References 25 publications

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“…The HWM is known as a method with simple implementation since it is based on the simplest wavelet [10]. Recently, the HWM was applied with success for solving Bratu-type equations [21] singularly perturbed differential equations with integral boundary conditions [22]. In [23][24][25][26], the HWM is combined with AI methods and tools.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of flight of the fragments with higher order Haar wavelet method

Kivistik,

Mehrparvar,

Eerme

et al. 2024

PEAS

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Fragments that have an irregular shape and move at high speeds are difficult to assess since experiments require high-tech solutions, and the differential equations that describe the motion cannot be solved analytically. Different numerical and function approximation methods are used to find the trajectory model. This work uses a state-of-the-art, higher order Haar wavelet method to approximate the trajectory model with empirically determined drag force. The initial conditions of the flight of the fragments are determined by the finite element method. The results obtained by utilizing the Haar wavelet method and the higher order Haar wavelet method are compared. The higher order Haar wavelet method outperforms the Haar wavelet method but allows for keeping the implementation complexity of the method in the same range. Utilizing the higher order Haar wavelet method leads to a reduction in the computational cost since the same accuracy with the Haar wavelet method can be achieved with the use of several order lower mesh.

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Section: Introductionmentioning

confidence: 99%

Dynamics of flight of the fragments with higher order Haar wavelet method

Kivistik,

Mehrparvar,

Eerme

et al. 2024

PEAS

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“…These problems have been broadly studied in the literature (see [17]). Ahsan et al [18] considered a Haar wavelet multi-resolution collocation method for singularly perturbed differential equations with integral boundary conditions. Cakir and Gunes [19] proposed an exponentially fitted difference scheme for singularly perturbed mixed integro-differential equations.…”

Section: Introductionmentioning

confidence: 99%

A Robust Numerical Method for a Singularly Perturbed Semilinear Problem with Integral Boundary Conditions

Temel,

Cakir

2024

Contemp. Math.

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In the present study, we provide an efficient numerical approach for solving singularly perturbed nonlinear ordinary differential equations with two integral boundary conditions. We specifically propose a numerical approach for the solution of a nonlinear singular perturbed problem with integral boundary conditions. To solve the nonlinear singularly perturbed issue, we also apply finite difference methods. It explores how a specific derivative and a problemsolving approach behave. Finally, a numerical technique that employs a finite difference scheme is built using a nonuniform mesh.

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“…The HCMHW for particular type of PDE called as regularized long wave equation is presented in [30]. In 2022, M. Ahsan et al extend the HCMHW presented in [44] for the solution of singularly perturbed nonlinear ODEs by utilizing the iterative quasi-linearization technique [47]. The HHWCM is also used to study the static response and buckling loads of multilayered composite beams [48] and vibration analysis of different types of beams [49][50][51].…”

Section: Introductionmentioning

confidence: 99%

A higher-order collocation method based on Haar wavelets for integro-differential equations with two-point integral condition

Ahsan,

Lei,

Alwuthaynani

et al. 2023

Phys. Scr.

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In this article, the higher-order Haar wavelet collocation method (HCMHW) is investigated to solve linear and nonlinear integro-differential equations (IDEs) with two types of conditions: simple initial condition and the point integral condition. We reproduce and compare the numerical results of the conventional Haar wavelet collocation method (CMHW) with those of HCMHW, demonstrating the superior performance of HCMHW across various conditions. Both methods effectively handle different types of given conditions. However, numerical results reveal that HCMHW exhibits a faster convergence rate than CMHW. To address nonlinear IDEs, we employ the quasi-linearization technique. The computational stability of both methods is evaluated through various experiments. Additionally, the article provides examples to illustrate the overall performance and accuracy of HCMHW compared to CMHW for both linear and nonlinear IDEs.

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A Haar wavelet multi-resolution collocation method for singularly perturbed differential equations with integral boundary conditions

Cited by 20 publications

References 25 publications

Dynamics of flight of the fragments with higher order Haar wavelet method

Dynamics of flight of the fragments with higher order Haar wavelet method

A Robust Numerical Method for a Singularly Perturbed Semilinear Problem with Integral Boundary Conditions

A higher-order collocation method based on Haar wavelets for integro-differential equations with two-point integral condition

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