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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 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.
3
Content available remote Comparison of modified ADM and classical finite difference method for some third-order and fifth-order KdV equations
Appadu Appanah Rao, Kelil Abey Sherif
Demonstratio Mathematica
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2021
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Vol. 54, nr 1
377--409
EN
The KdV equation, which appears as an asymptotic model in physical systems ranging from water waves to plasma physics, has been studied. In this paper, we are concerned with dispersive nonlinear KdV equations by using two reliable methods: Shehu Adomian decomposition method (STADM) and the classical finite difference method for solving three numerical experiments. STADM is constructed by combining Shehu’s transform and Adomian decomposition method, and the nonlinear terms can be easily handled using Adomian’s polynomials. The Shehu transform is used to accelerate the convergence of the solution series in most cases and to overcome the deficiency that is mainly caused by unsatisfied conditions in other analytical techniques. We compare the approximate and numerical results with the exact solution for the two numerical experiments. The third numerical experiment does not have an exact solution and we compare profiles from the two methods vs the space domain at some values of time. This study provides us with information about which of the two methods are effective based on the numerical experiment chosen. Knowledge acquired will enable us to construct methods for other related partial differential equations such as stochastic Korteweg-de Vries (KdV), KdV-Burgers, and fractional KdV equations.
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 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.
6
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.
7
Content available remote Application of Adomian decomposition method in analysis of settlement of random soil medium
Przewłócki J.
Archives of Civil Engineering
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2003
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Vol. 49, nr 2
191-212
EN
The paper presents the Adomian decomposition method and its application to solve a problem of random soil medium. Considered is a weighty layer of soil, founded on a rigid subsoil, on which a uniformly distributed vertical external load is acting. Analysis is carried out in uni-axial state of deformation. Elastic medium is assumed and its parameters are random functions. The problem is described by second order stochastic differential equation with random coefficient. Adomian decomposition method allowed to obtain approximate analytical solution with the assessment of its validity. Analysis of evaluation error was carried out for the expected value and the variance of a solution.
PL
Przedstawiono metodę dekompozycji Adomiana oraz jej zastosowanie do rozwiązania zagadnienia losowego ośrodka gruntowego. Rozpatrzono ważką warstwę gruntu, posadowioną na nieodkształcalnym podłożu, na którą działa równomiernie rozłożone pionowe obciążenie zewnętrzne. Analizę przeprowadzono w jednoosiowym stanie odkształcenia. Przyjęto, że ośrodek jest sprężysty, a jego parametry są funkcjami losowymi. Zagadnienie opisane jest poprzez stochastyczne równanie różniczkowe drugiego rzędu z losowym współczynnikiem. Metoda dekompozycji Adomiana pozwoliła na uzyskanie przybliżonego rozwiązania analitycznego wraz z oceną zakresu jego ważności. Przeprowadzono analizę błędu oszacowania dla wartości oczekiwanej i wariancji rozwiązania.
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