Fractional diffusion-wave equations: Hidden regularity for weak solutions

2021 ◽  
Vol 24 (4) ◽  
pp. 1015-1034
Author(s):  
Paola Loreti ◽  
Daniela Sforza
Keyword(s):  
Fractional Derivative ◽  
Weak Solutions ◽  
Caputo Fractional Derivative ◽  
Wave Equations ◽  
Regularity Result ◽  
Fractional Diffusion ◽  
Interpolation Spaces ◽  
Diffusion Wave ◽  
Regularity Properties

Abstract We prove a “hidden” regularity result for weak solutions of time fractional diffusion-wave equations where the Caputo fractional derivative is of order α ∈ (1, 2). To establish such result we analyse the regularity properties of the weak solutions in suitable interpolation spaces.

An Explicit Difference Method for Solving Fractional Diffusion and Diffusion-Wave Equations in the Caputo Form

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Joaquín Quintana Murillo ◽  
Santos Bravo Yuste
Keyword(s):  
Fractional Derivative ◽  
Difference Method ◽  
Wave Equations ◽  
Time Derivative ◽  
Fourier Method ◽  
Its Region ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
Von Neumann ◽  
The Stability

An explicit difference method is considered for solving fractional diffusion and fractional diffusion-wave equations where the time derivative is a fractional derivative in the Caputo form. For the fractional diffusion equation, the L1 discretization formula of the fractional derivative is employed, whereas the L2 discretization formula is used for the fractional diffusion-wave equation. In both equations, the spatial derivative is approximated by means of the three-point centered formula. The accuracy of the present method is similar to other well-known explicit difference schemes, but its region of stability is larger. The stability analysis is carried out by means of a kind of fractional von Neumann (or Fourier) method. The stability bound so obtained, which is given in terms of the Riemann zeta function, is checked numerically.

scholarly journals Nonexistence of Global Weak Solutions of a System of Nonlinear Wave Equations with Nonlinear Fractional Damping

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Mohamed Jleli ◽  
Mokhtar Kirane ◽  
Bessem Samet
Keyword(s):  
Weak Solution ◽  
Fractional Derivative ◽  
Nonlinear Wave ◽  
Weak Solutions ◽  
Caputo Fractional Derivative ◽  
Wave Equations ◽  
Global Weak Solution ◽  
Nonlinear Wave Equations ◽  
Global Weak Solutions ◽  
Fractional Damping

We consider the system of nonlinear wave equations with nonlinear time fractional damping utt+−Δmu+CD0,tαtσuq=vp,t>0,x∈ℝN,vtt+−Δmv+CD0,tβtδvr=vs,t>0,x∈ℝN,u0,x,ut0,x=u0x,u1x,x∈ℝN,u0,x,ut0,x=u0x,u1x,x∈ℝN,where u,v=ut,x,vt,x, m and N are positive natural numbers, p,q,r,s>1, σ,δ≥0, 0<α,β<1, and  CD0,tκ, 0<κ<1, is the Caputo fractional derivative of order κ. Namely, sufficient criteria are derived so that the system admits no global weak solution. To the best of our knowledge, the considered system was not previously studied in the literature.

scholarly journals FUNGSI WRIGHT SEBAGAI SOLUSI ANALITIK PERSAMAAN DIFUSI-GELOMBANG FRAKSIONAL PADA MEDIA VISKOELASTIS

2020 ◽  
Vol 3 (1) ◽  
pp. 19-33
Author(s):  
Ray Novita Yasa ◽  
Agus Yodi Gunawan
Keyword(s):  
Wave Equation ◽  
Analytical Solution ◽  
Fractional Derivative ◽  
Wave Equations ◽  
Equations Of Motion ◽  
Fractional Diffusion ◽  
Form Solution ◽  
Wright Function ◽  
Diffusion Wave ◽  
Viscoelastic Media

A fractional diffusion-wave equations in a fractional viscoelastic media can be constructed by using equations of motion and kinematic equations of viscoelasticmaterial in fractional order. This article concerns the fractional diffusion-wave equations in the fractional viscoelastic media for semi-infinite regions that satisfies signalling boundary value problems. Fractional derivative was used in Caputo sense. The analytical solution of the fractional diffusion-wave equation in the fractional viscoelastic media was solved by means of Laplace transform techniques in the term of Wright function for simple form solution. For general parameters, Numerical Inverse Laplace Transforms (NILT) was used to determine the solution.

Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder

2011 ◽  
Vol 14 (3) ◽  
Author(s):  
Yuriy Povstenko
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Neumann Boundary ◽  
Integral Transforms ◽  
Fractional Diffusion ◽  
Infinite Cylinder ◽  
Diffusion Wave ◽  
Spatial Coordinates ◽  
Axisymmetric Solutions

AbstractThe time-fractional diffusion-wave equation is considered in an infinite cylinder in the case of three spatial coordinates r, ϕ and z. The Caputo fractional derivative of the order 0 < α ≤ 2 is used. Several examples of problems with Dirichlet and Neumann boundary conditions at a surface of the cylinder are solved using the integral transforms technique. Numerical results are illustrated graphically.

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
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.

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

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.

scholarly journals A novel series method for fractional diffusion equation within Caputo fractional derivative

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 695-699 ◽  
Author(s):  
Sheng-Ping Yan ◽  
Wei-Ping Zhong ◽  
Xiao-Jun Yang
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Series Solution ◽  
Expansion Method ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Series Method ◽  
Time Fractional Diffusion Equation ◽  
Series Expansion Method

In this paper, we suggest the series expansion method for finding the series solution for the time-fractional diffusion equation involving Caputo fractional derivative.

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