In this paper, two methods are used to solve a nonlocal Cauchy problem of a delay differential eq... more In this paper, two methods are used to solve a nonlocal Cauchy problem of a delay differential equation; Adomian decomposition method (ADM) and Picard method. The existence and uniqueness of the solution are proved. The convergence of the series solution and the error analysis are studied.
In this paper, we apply the Adomian decomposition method (ADM) to solve Abel integral equation of... more In this paper, we apply the Adomian decomposition method (ADM) to solve Abel integral equation of the first and second kind. Abel integral equation is one the most important equations which appear in a lot of applications.
In this research, we employ a newly developed strategy based on a modified version of the Adomian... more In this research, we employ a newly developed strategy based on a modified version of the Adomian decomposition method (ADM) to solve nonlinear fractional differential equations (FDE) with both differential and nondifferential variables. FDE have disturbed the interest of many researchers. This is due to the development of both the theory and applications of fractional calculus. This track from various areas of fractional differential equations can be used to model various fields of science and engineering such as fluid flows, viscoelasticity, electrochemistry, control, electromagnetic, and many others. Several fractional derivative definitions have been presented, including Riemann-Liouville, Caputo,and Caputo-Fabrizio fractional derivative. We just need to calculate the first Adomain polynomial in this technique avoiding the hurdles in the nondifferentiable nonlinear terms' remaining polynomials. Furthermore, the proposed technique is easy to programme and produces the desired output with minimal work and time on the same processor. When compared to the exact solution, this method has the advantage of reducing calculation steps, while producing accurate results. The supporting evidence proves that modified Adomian decomposition has an advantage over traditional Adomian decomposition method which can be explained very clear with nonlinear fractional differential equations. Our computational examples with difficult issues are used to prove the new algorithm's efficiency. The results show that the modified ADM is powerful, which has a faster convergence solution than the original one. Convergence analysis is discussed, also the uniqueness is explained.
International Journal of Systems Science and Applied Mathematics, 2021
In this paper, Adomian decomposition method (ADM) will apply to solve nonlinear fractional differ... more In this paper, Adomian decomposition method (ADM) will apply to solve nonlinear fractional differential equations (FDEs) of Caputo sense. These type of equations is very important in engineering applications such as electrical networks, fluid flow, control theory and fractals theory. ADM give analytical solution in form of series solution so the convergence of the series solution and the error analysis will discuss. In addition, existence and uniqueness of the solution will prove. Some numerical examples will solve to test the validity of the method and the given theorems. A comparison of ADM solution with exact and numerical methods are given.
In this paper, we apply the Adomian decomposition method (ADM) for solving nonlinear system of no... more In this paper, we apply the Adomian decomposition method (ADM) for solving nonlinear system of nonlinear fractional differential equations (FDEs) of Riemann-Liouville sense. The existence and uniqueness of the solution will prove. The convergence of the series solution and the error analysis will discuss. Some numerical examples will solve to test the validity of the method.
In this paper, the Adomian decomposition method (ADM) and Picard technique are used to solve a cl... more In this paper, the Adomian decomposition method (ADM) and Picard technique are used to solve a class of nonlinear multidimensional fractional differential equations with Caputo-Fabrizio fractional derivative. The main advantage of the Caputo-Fabrizio fractional derivative appears in its non-singular kernel of a convolution type. The sufficient condition that guarantees a unique solution is obtained, the convergence of the series solution is discussed, and the maximum absolute error is estimated. Several numerical problems with an unknown exact solution are solved using the two techniques. A comparative study between the two solutions is presented. A comparative study shows that the time consumed by ADM is much smaller compared with the Picard technique.
We are concerned here with a nonlinear multidimensional fractional differential equation (FDE). T... more We are concerned here with a nonlinear multidimensional fractional differential equation (FDE). The existence of a unique solution will be proved. Convergence analysis of Adomian decompo-sition method (ADM) applied to these type of equations is discussed. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of Adomian's series solution. An application of such equations is Bagley–Torvik equation.
Journal of Applied Mathematics and Computing, 2009
ABSTRACT We are concerned here with a nonlinear multi-term fractional differential equation (FDE)... more ABSTRACT We are concerned here with a nonlinear multi-term fractional differential equation (FDE). The existence of a unique solution will be proved. Convergence analysis of Adomian decomposition method (ADM) applied to these type of equations is discussed. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of Adomian’s series solution. Some numerical examples are given, their ADM solutions are compared with a numerical method solutions. This numerical method is introduced in Podlubny (Fractional Differential Equations, Chap. 8, Academic Press, San Diego, 1999).
The comparison between the classical method of successive approximations (Picard) method and Adom... more The comparison between the classical method of successive approximations (Picard) method and Adomian decomposition method was studied in many papers for example ([15] and [37]). In this paper we are concerning with two analytical methods; the classical method of successive approximations (Picard) [18] and Adomian decomposition methods ([1]-[6], [16] and [17]) for a coupled system of quadratic integral equations of fractional order. Also, the existence and uniqueness of the solution and the convergence will be discussed for each method and some examples will be studied.
We are concerned here with a nonlinear quadratic integral equation (QIE) of Volterra type. The ex... more We are concerned here with a nonlinear quadratic integral equation (QIE) of Volterra type. The existence of a unique solution will be proved. Convergence analysis of Adomian decomposition method (ADM) applied to these type of equations is discussed and the maximum absolute truncated error of Adomian's series solution is estimated. Sometimes, when ADM is used, it produces di¢ cult integrals so, a numerical implementation technique (NIT) is used to overcome this problem.
In this paper, two methods are used to solve a nonlocal Cauchy problem of a delay differential eq... more In this paper, two methods are used to solve a nonlocal Cauchy problem of a delay differential equation; Adomian decomposition method (ADM) and Picard method. The existence and uniqueness of the solution are proved. The convergence of the series solution and the error analysis are studied.
In this paper, we apply the Adomian decomposition method (ADM) to solve Abel integral equation of... more In this paper, we apply the Adomian decomposition method (ADM) to solve Abel integral equation of the first and second kind. Abel integral equation is one the most important equations which appear in a lot of applications.
In this research, we employ a newly developed strategy based on a modified version of the Adomian... more In this research, we employ a newly developed strategy based on a modified version of the Adomian decomposition method (ADM) to solve nonlinear fractional differential equations (FDE) with both differential and nondifferential variables. FDE have disturbed the interest of many researchers. This is due to the development of both the theory and applications of fractional calculus. This track from various areas of fractional differential equations can be used to model various fields of science and engineering such as fluid flows, viscoelasticity, electrochemistry, control, electromagnetic, and many others. Several fractional derivative definitions have been presented, including Riemann-Liouville, Caputo,and Caputo-Fabrizio fractional derivative. We just need to calculate the first Adomain polynomial in this technique avoiding the hurdles in the nondifferentiable nonlinear terms' remaining polynomials. Furthermore, the proposed technique is easy to programme and produces the desired output with minimal work and time on the same processor. When compared to the exact solution, this method has the advantage of reducing calculation steps, while producing accurate results. The supporting evidence proves that modified Adomian decomposition has an advantage over traditional Adomian decomposition method which can be explained very clear with nonlinear fractional differential equations. Our computational examples with difficult issues are used to prove the new algorithm's efficiency. The results show that the modified ADM is powerful, which has a faster convergence solution than the original one. Convergence analysis is discussed, also the uniqueness is explained.
International Journal of Systems Science and Applied Mathematics, 2021
In this paper, Adomian decomposition method (ADM) will apply to solve nonlinear fractional differ... more In this paper, Adomian decomposition method (ADM) will apply to solve nonlinear fractional differential equations (FDEs) of Caputo sense. These type of equations is very important in engineering applications such as electrical networks, fluid flow, control theory and fractals theory. ADM give analytical solution in form of series solution so the convergence of the series solution and the error analysis will discuss. In addition, existence and uniqueness of the solution will prove. Some numerical examples will solve to test the validity of the method and the given theorems. A comparison of ADM solution with exact and numerical methods are given.
In this paper, we apply the Adomian decomposition method (ADM) for solving nonlinear system of no... more In this paper, we apply the Adomian decomposition method (ADM) for solving nonlinear system of nonlinear fractional differential equations (FDEs) of Riemann-Liouville sense. The existence and uniqueness of the solution will prove. The convergence of the series solution and the error analysis will discuss. Some numerical examples will solve to test the validity of the method.
In this paper, the Adomian decomposition method (ADM) and Picard technique are used to solve a cl... more In this paper, the Adomian decomposition method (ADM) and Picard technique are used to solve a class of nonlinear multidimensional fractional differential equations with Caputo-Fabrizio fractional derivative. The main advantage of the Caputo-Fabrizio fractional derivative appears in its non-singular kernel of a convolution type. The sufficient condition that guarantees a unique solution is obtained, the convergence of the series solution is discussed, and the maximum absolute error is estimated. Several numerical problems with an unknown exact solution are solved using the two techniques. A comparative study between the two solutions is presented. A comparative study shows that the time consumed by ADM is much smaller compared with the Picard technique.
We are concerned here with a nonlinear multidimensional fractional differential equation (FDE). T... more We are concerned here with a nonlinear multidimensional fractional differential equation (FDE). The existence of a unique solution will be proved. Convergence analysis of Adomian decompo-sition method (ADM) applied to these type of equations is discussed. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of Adomian's series solution. An application of such equations is Bagley–Torvik equation.
Journal of Applied Mathematics and Computing, 2009
ABSTRACT We are concerned here with a nonlinear multi-term fractional differential equation (FDE)... more ABSTRACT We are concerned here with a nonlinear multi-term fractional differential equation (FDE). The existence of a unique solution will be proved. Convergence analysis of Adomian decomposition method (ADM) applied to these type of equations is discussed. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of Adomian’s series solution. Some numerical examples are given, their ADM solutions are compared with a numerical method solutions. This numerical method is introduced in Podlubny (Fractional Differential Equations, Chap. 8, Academic Press, San Diego, 1999).
The comparison between the classical method of successive approximations (Picard) method and Adom... more The comparison between the classical method of successive approximations (Picard) method and Adomian decomposition method was studied in many papers for example ([15] and [37]). In this paper we are concerning with two analytical methods; the classical method of successive approximations (Picard) [18] and Adomian decomposition methods ([1]-[6], [16] and [17]) for a coupled system of quadratic integral equations of fractional order. Also, the existence and uniqueness of the solution and the convergence will be discussed for each method and some examples will be studied.
We are concerned here with a nonlinear quadratic integral equation (QIE) of Volterra type. The ex... more We are concerned here with a nonlinear quadratic integral equation (QIE) of Volterra type. The existence of a unique solution will be proved. Convergence analysis of Adomian decomposition method (ADM) applied to these type of equations is discussed and the maximum absolute truncated error of Adomian's series solution is estimated. Sometimes, when ADM is used, it produces di¢ cult integrals so, a numerical implementation technique (NIT) is used to overcome this problem.
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