Solving differential equations of fractional order using an optimization technique based on training artificial neural network

Pakdaman, Morteza; Ahmadian, Ali; Effati, Sohrab; Salahshour, Soheil; Băleanu, Dumitru

doi:10.1016/j.amc.2016.07.021

Cited by 98 publications

(67 citation statements)

References 39 publications

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“…Moreover, Figure B shows very good matching between the exact solution ( α = 1.0), the proposed optimized solutions, and the predictor‐corrector (PC) method published in Odibat and Momani for different values of α = 0.4, 0.6, 0.8, and 1.0 in the interval t ∈ [0, 5]. In addition, Table shows a comparison between the proposed results with the results reported in three recent papers for α = 0.4 and 0.8 in the interval t ∈ [0, 1]. Therefore, the proposed method provides similar or better accuracy results than the methods published before in previous studies …”

Section: Numerical Simulationssupporting

confidence: 58%

“…Example 1: Consider the following fractional‐order differential equation

D_{c}^{α} u ((), t) + u ((), t) = 0, u ((), 0) = 1, 0 < α \leq 1 .

The exact solution of this equation is given by

u ((), t) = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} t^{αk}}{normalΓ ((), italicαk + 1)} .

By applying the proposed method over t ∈ [0, 1], the used approximate solution based on the first four terms of the fractional‐order Chebyshev polynomials is given as follows:

u_{3} ((), t) = c_{0} + \sum_{i = 1}^{3} c_{i} i \sum_{k = 0}^{i} {(- 1)}^{i - k} \frac{((), i + k - 1)! 2^{2 k}}{((), i - k)! ((), 2 k)!} t^{βk},

and the residual error R ( t j ) for fractional order differential equation is given by

R ((), t_{j}) = \sum_{i = 0}^{3} c_{i} i \sum_{k = 1}^{i} {(- 1)}^{i - k} \frac{((), i + k - 1)! 2^{2 k}}{((), i - k)! ((), 2 k)!} \frac{italicΓ ((), k + 1)}{italicΓ ((), k + 1 - α)} {t_{j}}^{italicβk - α} + \sum_{i = 0}^{3} c_{i} i

…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…From Table , the maximum absolute error when choosing the value of β = α is less than when β = 1 (conventional case), which describes one of the advantages for the extra degree fractional‐order parameter. A comparison between the obtained results with the optimization method published in Pakdaman et al is added to Table where a percentage of reduction 97% has been achieved for the three discussed cases when α = β . To discuss the effect of number of terms ( M ) on the approximated solution, the previous process has been repeated for three different cases when M = 3, 4, and 5 as shown in Figure for the three fractional‐order scenarios.…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…The exact solution of this problem for α ( t ) = 1 is

u ((), t) = \frac{- 2}{t + 1}

. This problem has been solved by numerical methods in Pakdaman et al and Odibat and Momani when α ( t ) is constant value between 0 and 1. By applying the proposed method using 15 terms fractional‐order Chebyshev polynomials when β = α and over t ∈ [0, 5].…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…Therefore, the numerical methods for these equations have undergone fast growth in recent years. These methods are based on an optimization technique, Laplace transforms, spectral techniques, and iterative techniques . Other techniques are based on the operational matrix of fractional calculus of orthogonal polynomials with integer orders (Lucas polynomials, Legendre polynomials, and other) .…”

Section: Introductionmentioning

confidence: 99%

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The minimax approach for a class of variable order fractional differential equation

Semary

Hassan

Radwan

2019

Math Methods in App Sciences

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This paper introduces an approximate solution for Liouville‐Caputo variable order fractional differential equations with order 0 < α(t) ≤ 1. The solution is adapted using a family of fractional‐order Chebyshev functions with unknown coefficients. These coefficients have been obtained by using an optimization approach based on minimax technique and the least pth optimization function. Several linear and nonlinear fractional‐order differential equations are discussed using the proposed technique for fixed and variable order fractional‐order derivatives. Moreover, the response of RC charging circuit with variable order fractional capacitor is studied for different cases. Several comparisons with related published techniques have been added to illustrate the accuracy of the proposed approach.

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Section: Numerical Simulationssupporting

confidence: 58%

“…Example 1: Consider the following fractional‐order differential equation

D_{c}^{α} u ((), t) + u ((), t) = 0, u ((), 0) = 1, 0 < α \leq 1 .

The exact solution of this equation is given by

u ((), t) = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} t^{αk}}{normalΓ ((), italicαk + 1)} .

By applying the proposed method over t ∈ [0, 1], the used approximate solution based on the first four terms of the fractional‐order Chebyshev polynomials is given as follows:

u_{3} ((), t) = c_{0} + \sum_{i = 1}^{3} c_{i} i \sum_{k = 0}^{i} {(- 1)}^{i - k} \frac{((), i + k - 1)! 2^{2 k}}{((), i - k)! ((), 2 k)!} t^{βk},

and the residual error R ( t j ) for fractional order differential equation is given by

R ((), t_{j}) = \sum_{i = 0}^{3} c_{i} i \sum_{k = 1}^{i} {(- 1)}^{i - k} \frac{((), i + k - 1)! 2^{2 k}}{((), i - k)! ((), 2 k)!} \frac{italicΓ ((), k + 1)}{italicΓ ((), k + 1 - α)} {t_{j}}^{italicβk - α} + \sum_{i = 0}^{3} c_{i} i

…”

Section: Numerical Simulationsmentioning

confidence: 99%

Section: Numerical Simulationsmentioning

confidence: 99%

“…The exact solution of this problem for α ( t ) = 1 is

u ((), t) = \frac{- 2}{t + 1}

Section: Numerical Simulationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The minimax approach for a class of variable order fractional differential equation

Semary

Hassan

Radwan

2019

Math Methods in App Sciences

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show abstract

A neural network‐based selective harmonic elimination scheme for five‐level inverter

Maamar

Helaimi

Taleb

et al. 2021

Circuit Theory & Apps

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This paper presents the implementation of selective harmonic elimination (SHE) in a five-level inverter structure using artificial neural networks (ANNs). SHE is an effective low-frequency modulation technique to eliminate selected harmonics and control multilevel converters. The use of ANN-SHE requires the calculation of the optimum values of switching angles via the solving system of nonlinear equations for the total harmonic distortion (THD) reduction, where the nonlinear equations are founded by the complex Fourier series analysis of the inverter output voltage. The procured switching angle values are directly implemented by a multilayer perceptron (MLP) algorithm without a lookup table. The ANN model is obtained by training the neural network (NN), taking the modulation index (M) as an input and approximating switching angles as an output. A thorough analysis was carried out to show the programming steps of the proposed ANN-based SHE using Matlab/ Simulink environment. A realized inverter prototype steered by the proposed ANN-based SHE was tested with various modulation indexes on a real-time mode using a digital signal processor (DSP) C2000 Delfino-TMS320F28379Dembedded board. A comparison between the simulation results and the experimental data is presented. The obtained results illustrate that the experimental results match the simulation closely, and the ANN model provides a fast and precise estimate of the switching angles for each value of the modulation index.

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Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

Rasanan

Bajalan

Parand

et al. 2019

Math Methods in App Sciences

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By the rapid growth of available data, providing data-driven solutions for nonlinear (fractional) dynamical systems becomes more important than before. In this paper, a new fractional neural network model that uses fractional order of Jacobi functions as its activation functions for one of the hidden layers is proposed to approximate the solution of fractional differential equations and fractional partial differential equations arising from mathematical modeling of cognitive-decision-making processes and several other scientific subjects. This neural network uses roots of Jacobi polynomials as the training dataset, and the Levenberg-Marquardt algorithm is chosen as the optimizer. The linear and nonlinear fractional dynamics are considered as test examples showing the effectiveness and applicability of the proposed neural network. The numerical results are compared with the obtained results of some other networks and numerical approaches such as meshless methods. Numerical experiments are presented confirming that the proposed model is accurate, fast, and feasible.

show abstract

Solving differential equations of fractional order using an optimization technique based on training artificial neural network

Cited by 98 publications

References 39 publications

The minimax approach for a class of variable order fractional differential equation

The minimax approach for a class of variable order fractional differential equation

A neural network‐based selective harmonic elimination scheme for five‐level inverter

Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

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