Orthonormal Bernstein polynomials for solving nonlinear variable‐order time fractional fourth‐order diffusion‐wave equation with nonsingular fractional derivative

2020 ◽  
Author(s):  
M. H. Heydari ◽  
Z. Avazzadeh
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Fourth Order ◽  
Bernstein Polynomials ◽  
Variable Order ◽  
Diffusion Wave

scholarly journals A numerical scheme to solve variable order diffusion-wave equations

2019 ◽  
Vol 23 (Suppl. 6) ◽  
pp. 2063-2071 ◽  
Author(s):  
Ganji Moallem ◽  
Hossein Jafari ◽  
Abdullahi Adem
Keyword(s):  
Collocation Method ◽  
Algebraic System ◽  
Numerical Scheme ◽  
Wave Equations ◽  
Bernstein Polynomials ◽  
Operational Matrices ◽  
Variable Order ◽  
Bernstein Basis ◽  
Diffusion Wave ◽  
Variable Order Derivative

In this work, we consider variable order diffusion-wave equations. We choose variable order derivative in the Caputo sense. First, we approximate the unknown functions and its derivatives using Bernstein basis. Then, we obtain operational matrices based on Bernstein polynomials. Finally, with the help of these operational matrices and collocation method, we can convert variable order diffusion-wave equations to an algebraic system. Few examples are given to demonstrate the accuracy and the competence of the presented technique.

scholarly journals Time-Fractional Diffusion-Wave Equation with Mass Absorption in a Sphere under Harmonic Impact

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 433 ◽  
Author(s):  
Bohdan Datsko ◽  
Igor Podlubny ◽  
Yuriy Povstenko
Keyword(s):  
Fourier Transform ◽  
Wave Equation ◽  
Fractional Derivative ◽  
Graphical Representation ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
Mass Absorption ◽  
The Laplace Transform ◽  
Time Fractional Diffusion Equation ◽  
Time Fractional Derivative

The time-fractional diffusion equation with mass absorption in a sphere is considered under harmonic impact on the surface of a sphere. The Caputo time-fractional derivative is used. The Laplace transform with respect to time and the finite sin-Fourier transform with respect to the spatial coordinate are employed. A graphical representation of the obtained analytical solution for different sets of the parameters including the order of fractional derivative is given.

scholarly journals Analysis of two Legendre spectral approximations for the variable-coefficient fractional diffusion-wave equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Wenping Chen ◽  
Shujuan Lü ◽  
Hu Chen ◽  
Lihua Jiang
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Variable Coefficient ◽  
Equivalent Form ◽  
Fractional Diffusion ◽  
Stability And Convergence ◽  
Diffusion Wave ◽  
Fully Discrete ◽  
Spectral Approximations ◽  
Time Fractional Derivative

Abstract In this paper, we solve the variable-coefficient fractional diffusion-wave equation in a bounded domain by the Legendre spectral method. The time fractional derivative is in the Caputo sense of order $\gamma \in (1,2)$ γ ∈ ( 1 , 2 ) . We propose two fully discrete schemes based on finite difference in temporal and Legendre spectral approximations in spatial discretization. For the first scheme, we discretize the time fractional derivative directly by the $L_{1}$ L 1 approximation coupled with the Crank–Nicolson technique. For the second scheme, we transform the equation into an equivalent form with respect to the Riemann–Liouville fractional integral operator. We give a rigorous analysis of the stability and convergence of the two fully discrete schemes. Numerical examples are carried out to verify the theoretical results.

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