Numerical solution of fractional integro-differential equations by a hybrid collocation method

Ma, Xiaohua; Huang, Chengming

doi:10.1016/j.amc.2012.12.072

Cited by 72 publications

(42 citation statements)

References 26 publications

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“…Also, in order to confirm whether the mentioned iterative technique leads to higher accuracy, the test cases are compared in simulation with the samples provided in 35 . Below, we use the formal specifications η = 0.03 and γ = 0.05.…”

Section: Numerical Simulation and Discussionmentioning

confidence: 99%

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Using ANNs Approach for Solving Fractional Order Volterra Integro-differential Equations

Jafarian¹,

Rostami²,

Golmankhaneh³

et al. 2017

IJCIS

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Indeed, interesting properties of artificial neural networks approach made this non-parametric model a powerful tool in solving various complicated mathematical problems. The current research attempts to produce an approximate polynomial solution for special type of fractional order Volterra integrodifferential equations. The present technique combines the neural networks approach with the power series method to introduce an efficient iterative technique. To do this, a multi-layer feed-forward neural architecture is depicted for constructing a power series of arbitrary degree. Combining the initial conditions with the resulted series gives us a suitable trial solution. Substituting this solution instead of the unknown function and employing the least mean square rule, converts the origin problem to an approximated unconstrained optimization problem. Subsequently, the resulting nonlinear minimization problem is solved iteratively using the neural networks approach. For this aim, a suitable three-layer feed-forward neural architecture is formed and trained using a back-propagation supervised learning algorithm which is based on the gradient descent rule. In other words, discretizing the differential domain with a classical rule produces some training rules. By importing these to designed architecture as input signals, the indicated learning algorithm can minimize the defined criterion function to achieve the solution series coefficients. Ultimately, the analysis is accompanied by two numerical examples to illustrate the ability of the method. Also, some comparisons are made between the present iterative approach and another traditional technique. The obtained results reveal that our method is very effective, and in these examples leads to the better approximations.

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Section: Numerical Simulation and Discussionmentioning

confidence: 99%

“…where the ones obtained from the hybrid collocation (HC) method 35 . The maximum absolute errors (∥.∥ ∞ ) between approximate and exact solutions are numerically compared in Table 2.…”

Section: The Index Of Iterations X Rmentioning

confidence: 99%

Using ANNs Approach for Solving Fractional Order Volterra Integro-differential Equations

Jafarian¹,

Rostami²,

Golmankhaneh³

et al. 2017

IJCIS

View full text Add to dashboard Cite

show abstract

“…By substituting (20) into (19) and by ignoring some unnecessary terms, for sufficiently large N we can write…”

Section: (A) Polynomial Approximation Error Boundsmentioning

confidence: 99%

“…In [12], Taylor expansion approach was presented for solving a class of linear FIDE's including those of Fredholm and Volterra types. In [20], authors was solved FIDE's by adopting Hybrid collocation method to an equivalent integral equation of convolution type. In [14], Chebyshev Pseudospectral method was implemented for solving the linear and nonlinear systems of FIDE's.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations

Mokhtary

2015

Journal of Computational and Applied Mathematics

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Please cite this article as: P. Mokhtary, Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations, Journal of Computational and Applied Mathematics (2014), http://dx. AbstractIn this paper, Legendre Collocation method, an easy-to-use variant of the spectral methods for the numerical solution of a class of fractional integro-differential equations(FIDE's), is researched. In order to obtain high order accuracy for the approximation, the integral term in the resulting equation is approximated by using Legendre Gauss quadrature formula. An efficient convergence analysis of the proposed method is given and rate of convergence is established in the L 2 -norm. Due to the fact that the solutions of FIDE's usually have a weak singularity at origin, we use a variable transformation to change the original equation into a new equation with a smooth solution. We prove that after this regularization technique, numerical solution of the new equation by adopting the Legendre collocation method has exponentially rate of convergence. Numerical results are presented which clarify the high accuracy of the proposed method.

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“…Although Examples 6 and 7 are systems of FDIEs, the method described here can be applied. Example 1 was solved in [46] by using a hybrid collocation method. For Example 1, the exact solution can be obtained using the present method.…”

Section: Illustrative Examplementioning

confidence: 99%

Numerical solution of nonlinear fractional integro-differential equations by hybrid functions

Mashayekhi

Razzaghi

2015

Engineering Analysis with Boundary Elements

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Numerical solution of fractional integro-differential equations by a hybrid collocation method

Cited by 72 publications

References 26 publications

Using ANNs Approach for Solving Fractional Order Volterra Integro-differential Equations

Using ANNs Approach for Solving Fractional Order Volterra Integro-differential Equations

Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations

Numerical solution of nonlinear fractional integro-differential equations by hybrid functions

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