scholarly journals Similarity solutions of fractional order heat equations with variable coefficients

2016 ◽  
Vol 17 (1) ◽  
pp. 245 ◽  
Author(s):  
A. Elsaid ◽  
M. S. Abdel Latif ◽  
M. Maneea
Keyword(s):  
Fractional Order ◽  
Variable Coefficients ◽  
Similarity Solutions ◽  
Heat Equations

scholarly journals A General Iteration Formula of VIM for Fractional Heat- and Wave-Like Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Fukang Yin ◽  
Junqiang Song ◽  
Xiaoqun Cao
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Lagrange Multiplier ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Variable Coefficients ◽  
Iteration Formula ◽  
Variational Iteration

A general iteration formula of variational iteration method (VIM) for fractional heat- and wave-like equations with variable coefficients is derived. Compared with previous work, the Lagrange multiplier of the method is identified in a more accurate way by employing Laplace’s transform of fractional order. The fractional derivative is considered in Jumarie’s sense. The results are more accurate than those obtained by classical VIM and the same as ADM. It is shown that the proposed iteration formula is efficient and simple.

scholarly journals Numerical Methods for Fractional Order Singular Partial Differential Equations with Variable Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Asma Ali Elbeleze ◽  
Adem Kılıçman ◽  
Bachok M. Taib
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Variational Iteration Method ◽  
Convergent Series ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

We implement relatively analytical methods, the homotopy perturbation method and the variational iteration method, for solving singular fractional partial differential equations of fractional order. The process of the methods which produce solutions in terms of convergent series is explained. The fractional derivatives are described in Caputo sense. Some examples are given to show the accurate and easily implemented of these methods even with the presence of singularities.

scholarly journals Solution of linear fractional order systems with variable coefficients

2020 ◽  
Vol 23 (3) ◽  
pp. 753-763
Author(s):  
Ivan Matychyn ◽  
Viktoriia Onyshchenko
Keyword(s):  
Linear Systems ◽  
Fractional Order ◽  
Initial Value Problem ◽  
Transition Matrix ◽  
State Transition ◽  
Variable Coefficients ◽  
Explicit Solutions ◽  
Fractional Order Systems ◽  
Initial Value ◽  
Theoretical Results

AbstractThe paper deals with the initial value problem for linear systems of FDEs with variable coefficients involving Riemann–Liouville derivatives. The technique of the generalized Peano–Baker series is used to obtain the state-transition matrix. Explicit solutions are derived both in the homogeneous and inhomogeneous case. The theoretical results are supported by an example.

scholarly journals A numerical approach based on $ n $-dimensional fractional M\”{u}ntz-Legendre polynomials for solving fractional-order cohomological equations with variable coefficients

2021 ◽  
Author(s):  
Akbar Dehghan Nezhad ◽  
Mina Moghaddam Zeabadi
Keyword(s):  
Fractional Order ◽  
Continuous Time ◽  
Legendre Polynomials ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Variable Coefficients ◽  
Algebraic Equations ◽  
Cohomological Equation ◽  
Liouville Derivative ◽  
Cohomological Equations

This research presents a numerical approach to obtain the approximate solution of the n-dimensional cohomological equations of fractional order in continuous-time dynamical systems. For this purpose, the $ n $-dimensional fractional M\”{u}ntz-Legendre polynomials (or n-DFMLPs) are introduced. The operational matrix of the fractional Riemann-Liouville derivative is constructed by employing n-DFMLPs. Our method transforms the cohomological equation of fractional order into a system of algebraic equations. Therefore, the solution of that system of algebraic equations is the solution of the associated cohomological equation. The error bound and convergence analysis of the applied method under the $ L^{2} $-norm is discussed. Some examples are considered and discussed to confirm the efficiency and accuracy of our method.

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