Analytical solution of fractional variable order differential equations

Malesza, Wiktor; Macias, Michał; Sierociuk, Dominik

doi:10.1016/j.cam.2018.08.035

Cited by 37 publications

(17 citation statements)

References 30 publications

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“…Various authors have attempted to prove the existence and uniqueness of the solutions to VO-FDEs. Malesza et al [63] presented an approach based on switching schemes that realize different types of VO operators to obtain closed-form solutions to specific VO-FDEs. The existence and uniqueness of the solution of a generalized VO-FDE has been addressed in [64][65][66][67] using standard techniques in analysis and, particularly, the Arzela-Ascoli theorem.…”

Section: (C) Solution Methods For Variable-order Fractional Differentmentioning

confidence: 99%

Applications of variable-order fractional operators: a review

2020

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Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.

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Section: (C) Solution Methods For Variable-order Fractional Differentmentioning

confidence: 99%

Applications of variable-order fractional operators: a review

2020

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“…The origins of fractional calculus date back to the beginnings of calculus itself. The idea started in a correspondence in 1695 between Leibniz and L'Hopital, where they discussed the possibility of raising the differential operator to the power of 1/2 [1], [2]. Fractional calculus is the study of noninteger order derivatives and integrals where the order can be rational, real, or even complex [1].…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulations and FPGA Implementations of Fractional-Order Systems Based on Product Integration Rules

et al. 2020

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Product integration (PI) rules are well known numerical techniques that are used to solve differential equations of integer and, recently, fractional orders. Due to the high memory dependency of the PI rules used in solving fractional-order systems (FOS), their hardware implementation is very difficult and resources-demanding. In this paper, modified versions of the PI rules are introduced to facilitate their digital implementations. The studied rules are PI rectangular, PI trapezoidal, and predict-evaluate-correctevaluate (PECE) rules. The three modified versions of the PI rules are validated using a benchmark system of differential equations for different sizes of the memory window to show the effect of the window size on the solution accuracy. Additionally, the three modified versions of the PI rules are used to simulate a novel fractional-order chaotic system (FOCS) where its bifurcation diagrams are discussed with the window length parameter. The chaotic system is then implemented on a Field Programmable Gate Array (FPGA). There are only a few trials in literature to implement the fractional PECE algorithm on FPGA, nevertheless, the proposed FPGA realization is compared with the most recent of these trials. The FPGA implementations of the three PI rules are made using Xilinx ISE 14.7 on Artix 7 kit. The achieved throughput are 1280 Mbits/sec for PI rectangular, 128.8 Mbits/sec for PI trapezoidal, and 129.12 Mbits/sec for fractional-order PECE.

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“…In the past several decades, there exist several results on the existence of the solution for constant-order fractional differential equation. [16][17][18][19][20][21][22][23][24][25] Moreover, the boundary value problem has become an important research topic in the area of constant order fractional differential equations. For example, Jiang et al 26 studied the two-point boundary value problems for fractional differential equation with causal operator by lower and upper solution method and the monotone iterative technique.…”

Section: Introductionmentioning

confidence: 99%

The Existence of the Extremal Solution for the Boundary Value Problems of Variable Fractional Order Differential Equation With Causal Operator

Jiang¹,

Guirao

Saeed

2020

Fractals

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In this study, the two-point boundary value problem is considered for the variable fractional order differential equation with causal operator. Under the definition of the Caputo-type variable fractional order operators, the necessary inequality and the existence results of the solution are obtained for the variable order fractional linear differential equations according to Arzela–Ascoli theorem. Then, based on the proposed existence results and the monotone iterative technique, the existence of the extremal solution is studied, and the relative results are obtained based on the lower and upper solution. Finally, an example is provided to illustrate the validity of the theoretical results.

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Analytical solution of fractional variable order differential equations

Cited by 37 publications

References 30 publications

Applications of variable-order fractional operators: a review

Applications of variable-order fractional operators: a review

Numerical Simulations and FPGA Implementations of Fractional-Order Systems Based on Product Integration Rules

The Existence of the Extremal Solution for the Boundary Value Problems of Variable Fractional Order Differential Equation With Causal Operator

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