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1
Content available Numerical solution of a fractional coupled system with the Caputo-Fabrizio fractional derivative
Mansouri Ikram, Bekkouche Mohammed Moumen, Ahmed Abdelaziz Azeb
Journal of Applied Mathematics and Computational Mechanics
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2023
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Vol. 22, nr 1
46--56
EN
Within this work, we discuss the existence of solutions for a coupled system of linear fractional differential equations involving Caputo-Fabrizio fractional orders. We prove the existence and uniqueness of the solution by using the Picard-Lindel ̈of method and fixed point theory. Also, to compute an approximate solution of problem, we utilize the Adomian decomposition method (ADM), as this method provides the solution in the form of a series such that the infinite series converge to the exact solution. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method.
2
Content available Solution of fractional heat-like and fractional wave-like equation by using modern strategy
Mohamed Zain El Abden, Hamza Amjad, Elzaki Tarig, Algolam Mohamed, Elhussein Shiraz
Acta Mechanica et Automatica
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2023
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Vol. 17, no. 3
372--380
EN
This paper introduces a novel form of the Adomian decomposition (ADM) method for solving fractional-order heat-like and wave-like equations with starting and boundary value problems. The derivations are provided in the sense of Caputo. In order to help understanding, the generalised formulation of the current approach is provided. Several numerical examples of fractional-order diffusion-wave equations (FDWEs) are solved using the suggested method in this context. In addition to examining the applicability of the suggested method to the solving of fractional-order heat-like and wave-like equations, a graphical depiction of the solutions to three instructive cases was constructed. Solution graphs were arrived at for integer and fractional-order problems. The derived and exact solutions to integer-order problems were found to be in excellent agreement. The subject of the present research endeavour is the convergence of fractional-order solutions. This strategy is considered to be the most successful way of addressing fractional-order initial-boundary value issues in science and engineering. This strategy is presented here.
3
Content available Solutions of Benjamin-Bona-Mahony, modified Camassa-Holm and Degasperis Procesi equations using an iterative method
Kumar Manoj
Journal of Applied Mathematics and Computational Mechanics
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2022
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Vol. 21, nr 3
59--72
EN
In the present paper, we solve the non-linear Benjamin-Bona-Mahony, modified Camassa-Holm, and Degasperis-Procesi equations using an iterative method introduced by Daftardar-Gejji and Jafari. Results are compared with those obtained by other iterative methods such as the Adomian decomposition method and homotopy perturbation method. It is observed that the proposed method is computationally inexpensive and yields more accurate solutions than the Adomian decomposition method and the homotopy perturbation method.
4
Content available A new numerical technique for solving fractional Bratu’s initial value problems in the Caputo and Caputo-Fabrizio sense
Khalouta Ali, Kadem Abdelouahab
Journal of Applied Mathematics and Computational Mechanics
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2020
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Vol. 19, nr 1
43--56
EN
The purpose of this paper is to propose a new numerical technique called the natural decomposition method (NDM) for solving fractional Bratu’s initial value problems (FBIVP) in the Caputo and Caputo-Fabrizio sense. The NDM is a combined form of the natural transform method and the Adomian decomposition method. The numerical example is provided in order to validate the efficiency and reliability of the proposed method. The obtained results reveal that the proposed method is a very efficient and simple tool for solving fractional differential equations.
5
Content available Analytical solution of the time fractional Fokker-Planck equation
Sutradhar T., Datta B. K., Bera R. K.
International Journal of Applied Mechanics and Engineering
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2014
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Vol. 19, no. 2
435--440
EN
A nonperturbative approximate analytic solution is derived for the time fractional Fokker-Planck (F-P) equation by using Adomian’s Decomposition Method (ADM). The solution is expressed in terms of Mittag-Leffler function. The present method performs extremely well in terms of accuracy, efficiency and simplicity.
6
Content available remote Analytic solution of time fractional nonlinear dynamic system by modified decomposition method
Sutradhar T., Datta B. K., Bera R. K.
International Journal of Applied Mechanics and Engineering
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2012
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Vol. 17, no. 4
1327-1337
EN
Analytical and numerical results are reported for an analytical approximate solution of a nonlinear dynamic system containing fractional derivative by a modified decomposition method. Comparison with the exact and numerical solution shows that the present method performs extremely well in terms of accuracy, efficiency and simplicity.
7
Content available remote Analytical solution of forced-convective boundary-layer flow over a flat plate
Mirgolbabaei H., Barari A., Ibsen L. B., Esfahani M. G.
Archives of Civil and Mechanical Engineering
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2010
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Vol. 10, no 2
43-51
EN
In this letter, the problem of forced convection heat transfer over a horizontal flat plate is investigated by employing the Adomian Decomposition Method (ADM). The series solution of the nonlinear differential equations governing on the problem is developed. Comparison between results obtained and those of numerical solution shows excellent agreement, illustrating the effectiveness of the method. The solution obtained by ADM gives an explicit expression of temperature distribution and velocity distribution over a flat plate.
PL
W artykule przedstawiono zastosowanie metody dekompozycji Adomiana do wymuszonego, konwekcyjnie przepływu ciepła w poziomej, płaskiej płycie. Rozwiązania nieliniowych równań różniczkowych opisujących zagadnienie poszukiwana w postaci szeregów Adomiana. Z porównania otrzymanych wyników z wynikami innych metod numerycznych wynika doskonała ich zgodność, która potwierdza skuteczność zastosowanej metody. Otrzymane rozwiązanie pozwoliło jednoznacznie wyznaczyć rozkład i prędkości mian temperatury w analizowanej płycie.
8
Content available remote Application of Adomian Decomposition Method and Variational Iteration Method for the solution of fractional Basset differential equation
Ray S. S.
International Journal of Applied Mechanics and Engineering
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2009
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Vol. 14, no 4
1163-1167
EN
In this paper, the Adomian decomposition method (ADM) and variational iteration method (VIM) are implemented to obtain an approximate solution for a fractional Basset equation with an arbitrary order […].The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The approximate solution obtained using the VIM is exactly the same and in good agreement to that obtained by using the ADM.
9
Content available remote Application of Adomian Decomposition Method and Variational Iteration Method for the solution of a fractional (arbitrary) order differential equation
Ray S. S., Bera R. K.
International Journal of Applied Mechanics and Engineering
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2009
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Vol. 14, no 4
1169-1173
EN
In this paper, the Adomian decomposition method (ADM) and variational iteration method (VIM) are implemented to obtain an approximate solution to a fractional differential equation with an arbitrary order […]. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions to different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The approximate solution obtained using the VIM is exactly the same and in good agreement as that obtained by using the ADM.
10
Content available remote Application of the Adomian decomposition method for solving the heat equation in the cast-mould heterogeneous domain
Grzymkowski R., Pleszczyński M., Słota D.
Archives of Foundry Engineering
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2009
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Vol. 9, iss. 4
57-62
EN
The paper is focused on a method for solving the heat equation in a cast-mould heterogeneous domain. The discussed method makes use of the Adomian decomposition method. The derived calculations prove the effectiveness of the method for solving such types of problems.
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