Solving Nonlinear Pdes Using the Higher Order Haar Wavelet Method on Nonuniform and Adaptive Grids

Ratas, Mart; Salupere, Andrus; Majak, Jüri

doi:10.3846/mma.2021.12920

Cited by 28 publications

(15 citation statements)

References 78 publications

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“…Recently, the higher order Haar wavelet method (HOHWM) was introduced in [13] in order to improve the accuracy and convergence of the previously proposed Haar wavelet method. The HOHWM has been applied with success to solving differential equations, vibration, and buckling response of beams [14][15][16][17][18]. Theoretical and numerical analyses of the free and forced vibration of homogeneous and functionally graded Timoshenko beams have been performed [19][20][21][22].…”

Section: Computational Mechanicsmentioning

confidence: 99%

Free vibration analysis of tapered Timoshenko beam with higher order Haar wavelet method

Arda¹,

Karjust²,

Majak³

et al. 2022

PEAS

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Section: Computational Mechanicsmentioning

confidence: 99%

Free vibration analysis of tapered Timoshenko beam with higher order Haar wavelet method

Arda¹,

Karjust²,

Majak³

et al. 2022

PEAS

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“…Various applications of HWCM in the approximation theory can be seen in [18][19][20][21][22][23][24][25][26][27]. Some of the recent work using Haar wavelets is given in [28][29][30][31][32][33][34][35][36].…”

Section: Introductionmentioning

confidence: 99%

A Modified Algorithm Based on Haar Wavelets for the Numerical Simulation of Interface Models

Rana

Asif

Haider

et al. 2022

Journal of Function Spaces

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In this paper, a new numerical technique is proposed for the simulations of advection-diffusion-reaction type elliptic and parabolic interface models. The proposed technique comprises of the Haar wavelet collocation method and the finite difference method. In this technique, the spatial derivative is approximated by truncated Haar wavelet series, while for temporal derivative, the finite difference formula is used. The diffusion coefficients, advection coefficients, and reaction coefficients are considered discontinuously across the fixed interface. The newly established numerical technique is applied to both linear and nonlinear benchmark interface models. In the case of linear interface models, Gauss elimination method is used, whereas for nonlinear interface models, the nonlinearity is removed by using the quasi-Newton linearization technique. The L ∞ errors are calculated for different number of collocation points. The obtained numerical results are compared with the immersed interface method. The stability and convergence of the method are also discussed. On the whole, the numerical results show more efficiency, better accuracy, and simpler applicability of the newly developed numerical technique compared to the existing methods in literature.

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“…A survey of some of the earlier work can be found in [32][33][34][35][36] and a review on harmonic wavelets is presented in [37]. Applications of Haar wavelet for numerical approximations are indicated in references [38][39][40][41][42][43][44][45][46][47][48][49][50]. Recently, Haar wavelet method has been extended to solve fractional partial differential equations [51].…”

Section: Introductionmentioning

confidence: 99%

On the numerical solution of some differential equations with nonlocal integral boundary conditions via Haar wavelet

Aziz¹,

Nisar²,

Islam³

2022

PEAS

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Differential equations with nonlocal boundary conditions are used to model a number of physical phenomena encountered in situations where data on the boundary cannot be measured directly. This study explores numerical solutions to elliptic, parabolic and hyperbolic equations with two different types of nonlocal integral boundary conditions. The numerical solutions are obtained using the Haar wavelet collocation method with the aid of Finite Differences for time derivatives. The method is applicable to both linear and nonlinear problems. To obtain the numerical solutions, Gauss elimination method is used for linear and Newton's method for nonlinear differential equations. The validity of the proposed method is demonstrated by solving several benchmark test problems from the literature: two elliptic linear and two nonlinear samples covering both types of nonlocal integral boundary conditions; one nonlinear and two linear test problems for parabolic partial differential equations; two linear samples for hyperbolic partial differential equations. The accuracy of the method is verified by comparing the numerical results with the analytical solutions. The numerical results confirm that the method is simple and effective.

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Solving Nonlinear Pdes Using the Higher Order Haar Wavelet Method on Nonuniform and Adaptive Grids

Cited by 28 publications

References 78 publications

Free vibration analysis of tapered Timoshenko beam with higher order Haar wavelet method

Free vibration analysis of tapered Timoshenko beam with higher order Haar wavelet method

A Modified Algorithm Based on Haar Wavelets for the Numerical Simulation of Interface Models

On the numerical solution of some differential equations with nonlocal integral boundary conditions via Haar wavelet

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