Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations

The implementation of numerical methods to solve and study equations for cardiac wave propagation in realistic geometries is very costly, in terms of computational resources. The aim of this work is to show the improvement that can be obtained with Chebyshev polynomials-based methods over the classical finite difference schemes to obtain numerical solutions of cardiac models. To this end, we present a Chebyshev multidomain (CMD) Pseudospectral method to solve a simple two variable cardiac models on three-dimensional anisotropic media and we show the usefulness of the method over the traditional finite differences scheme widely used in the literature.

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Using Fractional Bernoulli Wavelets for Solving Fractional Diffusion Wave Equations with Initial and Boundary Conditions

Fractal and Fractional ◽

10.3390/fractalfract5040212 ◽

2021 ◽

Vol 5 (4) ◽

pp. 212

Author(s):

Monireh Nosrati Sahlan ◽

Hojjat Afshari ◽

Jehad Alzabut ◽

Ghada Alobaidi

Keyword(s):

Fractional Order ◽

Fractional Derivatives ◽

Wave Equations ◽

Bernoulli Polynomials ◽

Fractional Diffusion ◽

Computational Time ◽

Operational Matrices ◽

Algebraic Equations ◽

Diffusion Wave ◽

Klein Gordon

In this paper, fractional-order Bernoulli wavelets based on the Bernoulli polynomials are constructed and applied to evaluate the numerical solution of the general form of Caputo fractional order diffusion wave equations. The operational matrices of ordinary and fractional derivatives for Bernoulli wavelets are set via fractional Riemann–Liouville integral operator. Then, these wavelets and their operational matrices are utilized to reduce the nonlinear fractional problem to a set of algebraic equations. For solving the obtained system of equations, Galerkin and collocation spectral methods are employed. To demonstrate the validity and applicability of the presented method, we offer five significant examples, including generalized Cattaneo diffusion wave and Klein–Gordon equations. The implementation of algorithms exposes high accuracy of the presented numerical method. The advantage of having compact support and orthogonality of these family of wavelets trigger having sparse operational matrices, which reduces the computational time and CPU requirements.

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Wave propagation, stress relaxation, and grain-to-grain shearing in saturated, unconsolidated marine sediments

The Journal of the Acoustical Society of America ◽

10.1121/1.1322018 ◽

2000 ◽

Vol 108 (6) ◽

pp. 2796-2815 ◽

Cited By ~ 117

Author(s):

Michael J. Buckingham

Keyword(s):

Wave Propagation ◽

Stress Relaxation ◽

Marine Sediments

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Analytic and Numerical Solutions of Space-Time Fractional Diffusion Wave Equations with Different Fractional Order

Computational Science – ICCS 2021 - Lecture Notes in Computer Science ◽

10.1007/978-3-030-77961-0_34 ◽

2021 ◽

pp. 408-421

Author(s):

Abhishek Kumar Singh ◽

Mani Mehra

Keyword(s):

Fractional Order ◽

Wave Equations ◽

Numerical Solutions ◽

Fractional Diffusion ◽

Space Time ◽

Diffusion Wave

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Robustness of Boundary Control of Fractional Wave Equations With Delayed Boundary Measurement Using Fractional Order Controller and the Smith Predictor

Volume 6: 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C ◽

10.1115/detc2005-85299 ◽

2005 ◽

Cited By ~ 2

Author(s):

Jinsong Liang ◽

Weiwei Zhang ◽

YangQuan Chen ◽

Igor Podlubny

Keyword(s):

Fractional Order ◽

Wave Equations ◽

Small Difference ◽

Smith Predictor ◽

Fractional Wave Equation ◽

Instability Problem ◽

Boundary Measurement ◽

Fractional Wave Equations ◽

Fractional Order Controller ◽

Better Than

In this paper, we analyze the robustness of the fractional wave equation with a fractional order boundary controller subject to delayed boundary measurement. Conditions are given to guarantee stability when the delay is small. For large delays, the Smith predictor is applied to solve the instability problem and the scheme is proved to be robust against a small difference between the assumed delay and the actual delay. The analysis shows that fractional order controllers are better than integer order controllers in the robustness against delays in the boundary measurement.

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