Subordination principle for fractional diffusion-wave equations of Sobolev type

2020 ◽  
Vol 23 (2) ◽  
pp. 427-449
Author(s):  
Rodrigo Ponce
Keyword(s):  
Banach Space ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Wave Equations ◽  
Mild Solutions ◽  
Fractional Diffusion ◽  
Sobolev Type ◽  
Diffusion Wave ◽  
Fractional Differential ◽  
Resolvent Families

AbstractIn this paper we study subordination principles for fractional differential equations of Sobolev type in Banach space. With the help of the theory of Sobolev type resolvent families (known also as propagation family) as well as these subordination principles, we obtain the existence of mild solutions for this kind of equations. We study simultaneously the case 0 < α < 1 and 1 < α < 2 for the Caputo and Riemann-Liouville fractional derivatives.

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 Sinc-Chebyshev Collocation Method for a Class of Fractional Diffusion-Wave Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Zhi Mao ◽  
Aiguo Xiao ◽  
Zuguo Yu ◽  
Long Shi
Keyword(s):  
Variable Coefficient ◽  
Fractional Derivatives ◽  
Wave Equations ◽  
Fractional Diffusion ◽  
Algebraic Equations ◽  
Chebyshev Collocation Method ◽  
Diffusion Wave ◽  
Linear Algebraic Equations ◽  
Collocation Technique ◽  
Sinc Functions

This paper is devoted to investigating the numerical solution for a class of fractional diffusion-wave equations with a variable coefficient where the fractional derivatives are described in the Caputo sense. The approach is based on the collocation technique where the shifted Chebyshev polynomials in time and the sinc functions in space are utilized, respectively. The problem is reduced to the solution of a system of linear algebraic equations. Through the numerical example, the procedure is tested and the efficiency of the proposed method is confirmed.

scholarly journals On solutions of linear fractional differential equations and systems thereof

2019 ◽  
Vol 22 (2) ◽  
pp. 479-494
Author(s):  
Khongorzul Dorjgotov ◽  
Hiroyuki Ochiai ◽  
Uuganbayar Zunderiya
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Wave Equations ◽  
Invariant Solutions ◽  
Diffusion Wave ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations

Abstract We derive exact solutions to classes of linear fractional differential equations and systems thereof expressed in terms of generalized Wright functions and Fox H-functions. These solutions are invariant solutions of diffusion-wave equations obtained through certain transformations, which are briefly discussed. We show that the solutions given in this work contain previously known results as particular cases.

scholarly journals Resolvent Positive Operators and Positive Fractional Resolvent Families

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Nan-Ding Li ◽  
Ru Liu ◽  
Miao Li
Keyword(s):  
Banach Space ◽  
Fractional Differential Equations ◽  
Ordered Banach Space ◽  
Resolvent Family ◽  
Completely Monotonic ◽  
Densely Defined Operator ◽  
Fractional Resolvent Families ◽  
Fractional Differential ◽  
Densely Defined ◽  
Resolvent Families

This paper is concerned with positive α -times resolvent families on an ordered Banach space E (with normal and generating cone), where 0 < α ≤ 2 . We show that a closed and densely defined operator A on E generates a positive exponentially bounded α -times resolvent family for some 0 < α < 1 if and only if, for some ω ∈ ℝ , when λ > ω , λ ∈ ρ A , R λ , A ≥ 0 and sup λ R λ , A : λ ≥ ω < ∞ . Moreover, we obtain that when 0 < α < 1 , a positive exponentially bounded α -times resolvent family is always analytic. While A generates a positive α -times resolvent family for some 1 < α ≤ 2 if and only if the operator λ α − 1 λ α − A − 1 is completely monotonic. By using such characterizations of positivity, we investigate the positivity-preserving of positive fractional resolvent family under positive perturbations. Some examples of positive solutions to fractional differential equations are presented to illustrate our results.

scholarly journals Analysis of Fractal Wave Equations by Local Fractional Fourier Series Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yong-Ju Yang ◽  
Dumitru Baleanu ◽  
Xiao-Jun Yang
Keyword(s):  
Fourier Series ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Present Method ◽  
Fractional Derivatives ◽  
Wave Equations ◽  
Series Method ◽  
Fourier Series Method ◽  
Fractional Differential

The fractal wave equations with local fractional derivatives are investigated in this paper. The analytical solutions are obtained by using local fractional Fourier series method. The present method is very efficient and accurate to process a class of local fractional differential equations.

Sign in / Sign up

Export Citation Format

Share Document