Efficient computational algorithms for solving one class of fractional boundary value problems

Irandoust‐pakchin, Safar; Kheiri, Hossein; Abdi-Mazraeh, Somayeh

doi:10.1134/s0965542513070117

Cited by 10 publications

(7 citation statements)

References 27 publications

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“…Remark In Nagy et al, Example 6.2 case III with n = 2 and l = 1 is solved. The numerical results obtained by Nagy et al are approximately coincident with the numerical results obtained by our suggested method. Case IV: (Bagely‐Torvik equation) Which arises in the modeling of the motion of a rigid plate immersed in Newtonian fluid, it obtained in case of

p = q = r = 1, u_{0} = u_{1} = 0, α (t) = 2, β (t) = \frac{3}{2}, and g (t) = t^{2} + 2 + \frac{4}{\sqrt{π}} t^{0.5}

The exact solution of Example 6.2 case IV is u ( t ) = t 2 . The three unknown coefficients are calculated by applying the suggested method constructed in Section with n = 2 in addition to the same steps followed in Example .…”

Section: Illustrative Examplessupporting

confidence: 71%

“…We obtain a great accuracy of the problem using the suggested method in this article because it yields the exact solution of the problem for very small terms of the shifted Legendre polynomials n = 2. Thus, the proposed method is more accurate than other methods given in Irandoust‐Pakchin et al Moreover, Figure presents the exact solution and approximate solutions for Example 6.2 case II in case of ( n = 10, l = 3) in addition to Example 6.2 case with the values ( n = l = 5). Figure proves and supports the idea of solving the multiterm fractional variable‐order differential equation using the suggested method for various choices of terms of series n and the interval lengths l .…”

Section: Illustrative Examplesmentioning

confidence: 81%

“…It is clear that we can obtain the exact solution of the Bagely‐Torvik equation using a few terms of the shifted Legendre polynomials n = 2. Although, the best results of the same problem obtained by Irandoust‐Pakchin are 6 × 10 −10 , 2 × 10 −10 , and the exact solution when n → ∞ . We obtain a great accuracy of the problem using the suggested method in this article because it yields the exact solution of the problem for very small terms of the shifted Legendre polynomials n = 2.…”

Section: Illustrative Examplesmentioning

confidence: 84%

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Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

El‐Sayed

Agarwal

2019

Math Methods in App Sciences

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In this paper, shifted Legendre polynomials will be used for constructing the numerical solution for a class of multiterm variable‐order fractional differential equations. In the proposed method, the shifted Legendre operational matrix of the fractional variable‐order derivatives will be investigated. The fundamental problem is reduced to an algebraic system of equations using the constructed matrix and the collocation technique, which can be solved numerically. The error estimate of the proposed method is investigated. Some numerical examples are presented to prove the applicability, generality, and accuracy of the suggested method.

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p = q = r = 1, u_{0} = u_{1} = 0, α (t) = 2, β (t) = \frac{3}{2}, and g (t) = t^{2} + 2 + \frac{4}{\sqrt{π}} t^{0.5}

Section: Illustrative Examplessupporting

confidence: 71%

Section: Illustrative Examplesmentioning

confidence: 81%

Section: Illustrative Examplesmentioning

confidence: 84%

See 1 more Smart Citation

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

El‐Sayed

Agarwal

2019

Math Methods in App Sciences

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“…Also, with a base size of 2, i.e., = 1, = 1, = 1, we obtain the exact solution. The paper 25 also obtains the exact solution, for base size 3 (n=2), and the paper 27 obtains the exact solution when → ∞.…”

Section: Examplementioning

confidence: 87%

A numerical method to solve multiterm fractional variable-order differential equation

Postavaru

Toma

2022

Preprint

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In this paper we build the fractional hybrid function of block-pulse functions and Fibonacci polynomials (FHBPF) to numerically solve a class of multiterm variable-order fractional differential equations. We consider fractional derivatives in the Caputo sense and fractional integrals in the Riemann-Liouville sense. We construct an exact integral operator attached to the FHBPF, based on the incomplete beta functions. With the help of the Newton-Cotes collocation procedure, we transform differential equations into systems of algebraic equations, which can be solved by traditional methods such as Newton’s iterative method. We also propose an error investigation method. We conclude the paper with some numerical examples to demonstrate the applicability of the method, but also its simplicity and accuracy.

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“…In recent decades, fractional delay differential equations have had an important role in engineering and natural sciences. Applications of these equations include hydrology, signal processing, control theory, medical sciences, networks, cell biology, climate models, infectious diseases, navigation prediction, circulating blood, population dynamics, oncolytic virotherapy, delayed plant disease model, the body reaction to carbon dioxide, and many others [3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

Numerical Approach for Solving the Fractional Pantograph Delay Differential Equations

Hajishafieiha

Abbasbandy

2022

Complexity

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A new class of polynomials investigates the numerical solution of the fractional pantograph delay ordinary differential equations. These polynomials are equipped with an auxiliary unknown parameter a , which is obtained using the collocation and least-squares methods. In this study, the numerical solution of the fractional pantograph delay differential equation is displayed in the truncated series form. The upper bound of the solution as well as the error analysis and the rate of convergence theorem are also investigated in this study. In five examples, the numerical results of the present method have been compared with other methods. For the first time, a -polynomials are used in this study to numerically solve delay equations, and accurate approximations have been displayed.

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Efficient computational algorithms for solving one class of fractional boundary value problems

Cited by 10 publications

References 27 publications

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

A numerical method to solve multiterm fractional variable-order differential equation

Numerical Approach for Solving the Fractional Pantograph Delay Differential Equations

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