MUSCL reconstruction and Haar wavelets

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

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Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽

10.3390/math7100923 ◽

2019 ◽

Vol 7 (10) ◽

pp. 923 ◽

Cited By ~ 1

Author(s):

Abdul Ghafoor ◽

Sirajul Haq ◽

Manzoor Hussain ◽

Poom Kumam ◽

Muhammad Asif Jan

Keyword(s):

Finite Difference ◽

Wave Equations ◽

Quadrature Rule ◽

Approximate Solutions ◽

Fractional Diffusion ◽

Haar Wavelets ◽

Test Problems ◽

Algebraic Equations ◽

Diffusion Wave ◽

Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

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Green–Haar wavelets method for generalized fractional differential equations

Advances in Difference Equations ◽

10.1186/s13662-020-02974-6 ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 1

Author(s):

Mujeeb ur Rehman ◽

Dumitru Baleanu ◽

Jehad Alzabut ◽

Muhammad Ismail ◽

Umer Saeed

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Haar Wavelet ◽

Haar Wavelets ◽

Computational Time ◽

Operational Matrices ◽

Numerical Techniques ◽

Fractional Differential ◽

Fractional Boundary Value Problems ◽

Better Than

Abstract The objective of this paper is to present two numerical techniques for solving generalized fractional differential equations. We develop Haar wavelets operational matrices to approximate the solution of generalized Caputo–Katugampola fractional differential equations. Moreover, we introduce Green–Haar approach for a family of generalized fractional boundary value problems and compare the method with the classical Haar wavelets technique. In the context of error analysis, an upper bound for error is established to show the convergence of the method. Results of numerical experiments have been documented in a tabular and graphical format to elaborate the accuracy and efficiency of addressed methods. Further, we conclude that accuracy-wise Green–Haar approach is better than the conventional Haar wavelets approach as it takes less computational time compared to the Haar wavelet method.

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