A new numerical technique for solving the local fractional diffusion equation
TL;DR: The numerical result presented here illustrates the efficiency and accuracy of the proposed computational technique in order to solve the partial differential equations involving local fractional derivatives.
About: This article is published in Applied Mathematics and Computation.The article was published on 2016-02-01. It has received 137 citations till now. The article focuses on the topics: Fractional calculus & First-order partial differential equation.
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TL;DR: The present methodology is shown to provide a useful approach to solve the local fractional nonlinear partial differential equations (LFNPDEs) in mathematical physics.
Abstract: In this paper, a family of local fractional two-dimensional Burgers-type equations (2DBEs) is investigated. The local fractional Riccati differential equation method is proposed here for the first time. The travelling wave transformation of the non-differentiable type is presented. The non-differentiable exact travelling wave solutions for the problems are obtained. The present methodology is shown to provide a useful approach to solve the local fractional nonlinear partial differential equations (LFNPDEs) in mathematical physics.
232 citations
TL;DR: In this paper, a new fractional derivative without singular kernel is proposed for modeling the steady state heat-conduction problem and the analytical solution of the fractional-order heat flow is also obtained by means of the Laplace transform.
Abstract: In this article we propose a new fractional derivative without singular
kernel. We consider the potential application for modeling the steady
heat-conduction problem. The analytical solution of the fractional-order heat
flow is also obtained by means of the Laplace transform.
195 citations
Cites background from "A new numerical technique for solvi..."
...Key words: heat conduction, steady heat flow, analytical solution, Laplace transform, fractional derivative without singular kernel Introduction Fractional derivatives with singular kernel [1], namely, the Riemann-Liouville [2-3], Caputo [4-5] and other derivatives (see [6-8] and the references therein), have nowadays a wide application in the field of heat-transfer engineering....
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TL;DR: The theoretical analysis and high-accuracy of the proposed method are verified, and Comparative results indicate that the accuracy of the new discretization technique is superior to the other methods available in the literature.
130 citations
TL;DR: New functions based on the Bernoulli wavelets are defined to obtain the numerical solution of fractional-order pantograph differential equations in a large interval to reduce the problem to a set of algebraic equations.
128 citations
TL;DR: A technique to discretize fractional differential equations with variable-order operators with resulting model and a particle swarm optimization algorithm are used to search for the optimal parameters of the VFPID controllers.
123 citations
References
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Book•
02 Mar 2006
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
Abstract: 1. Preliminaries. 2. Fractional Integrals and Fractional Derivatives. 3. Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems. 4. Methods for Explicitly solving Fractional Differential Equations. 5. Integral Transform Methods for Explicit Solutions to Fractional Differential Equations. 6. Partial Fractional Differential Equations. 7. Sequential Linear Differential Equations of Fractional Order. 8. Further Applications of Fractional Models. Bibliography Subject Index
11,492 citations
"A new numerical technique for solvi..." refers background in this paper
...Fractional derivative operators (FDOs) have successfully been observed to provide mathematical analytic tools with potential for applications in mathematical sciences, physics and mechanics (see [1,2])....
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Book•
24 Aug 2007
TL;DR: In the last two decades, fractional differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory and signal and image processing as discussed by the authors.
Abstract: In the last two decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory and signal and image processing. For example, in the last three fields, some important considerations such as modelling, curve fitting, filtering, pattern recognition, edge detection, identification, stability, controllability, observability and robustness are now linked to long-range dependence phenomena. Similar progress has been made in other fields listed here. The scope of the book is thus to present the state of the art in the study of fractional systems and the application of fractional differentiation. As this volume covers recent applications of fractional calculus, it will be of interest to engineers, scientists, and applied mathematicians.
1,119 citations
01 Jan 2006
1,096 citations
TL;DR: In this paper, the basic theory for the initial value problem of fractional differential equations involving Riemann-Liouville differential operators is discussed employing the classical approach, and the theory of inequalities, local existence, extremal solutions, comparison result and global existence of solutions are considered.
Abstract: In this paper, the basic theory for the initial value problem of fractional differential equations involving Riemann–Liouville differential operators is discussed employing the classical approach. The theory of inequalities, local existence, extremal solutions, comparison result and global existence of solutions are considered.
1,035 citations
Book•
09 Mar 2011
TL;DR: In this article, the Ginzburg-Landau Equation for Fractal Media and Fokker-Planck Equation of Fractal Distributions of Probability are presented.
Abstract: Fractional Continuous Models of Fractal Distributions.- Fractional Integration and Fractals.- Hydrodynamics of Fractal Media.- Fractal Rigid Body Dynamics.- Electrodynamics of Fractal Distributions of Charges and Fields.- Ginzburg-Landau Equation for Fractal Media.- Fokker-Planck Equation for Fractal Distributions of Probability.- Statistical Mechanics of Fractal Phase Space Distributions.- Fractional Dynamics and Long-Range Interactions.- Fractional Dynamics of Media with Long-Range Interaction.- Fractional Ginzburg-Landau Equation.- Psi-Series Approach to Fractional Equations.- Fractional Spatial Dynamics.- Fractional Vector Calculus.- Fractional Exterior Calculus and Fractional Differential Forms.- Fractional Dynamical Systems.- Fractional Calculus of Variations in Dynamics.- Fractional Statistical Mechanics.- Fractional Temporal Dynamics.- Fractional Temporal Electrodynamics.- Fractional Nonholonomic Dynamics.- Fractional Dynamics and Discrete Maps with Memory.- Fractional Quantum Dynamics.- Fractional Dynamics of Hamiltonian Quantum Systems.- Fractional Dynamics of Open Quantum Systems.- Quantum Analogs of Fractional Derivatives.
1,031 citations
"A new numerical technique for solvi..." refers background in this paper
...Fractional derivative operators (FDOs) have successfully been observed to provide mathematical analytic tools with potential for applications in mathematical sciences, physics and mechanics (see [1,2])....
[...]