scholarly journals APPLICATION OF ADOMIAN DECOMPOSITION METHOD, TAYLOR SERIES METHOD AND A VARIATIONAL ITERATIONS METHOD TO SOLVING A SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

2019 ◽  
Vol 73 (05) ◽  
pp. 1-5
Author(s):  
Ablakul Abdirashidov ◽  
◽  
Bekzod Ortikov ◽  
Nurshod Kadirov ◽  
Akmaljon Abdurashidov ◽  
...  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Taylor Series ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Second Order ◽  
Series Method ◽  
Adomian Decomposition ◽  
Taylor Series Method

scholarly journals Adomian decomposition method for solving initial value problems in vibration models

2018 ◽  
Vol 159 ◽  
pp. 02007
Author(s):  
Sudi Mungkasi ◽  
I Made Wicaksana Ekaputra
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
External Source ◽  
Accurate Solution ◽  
Second Order ◽  
Initial Value ◽  
Adomian Decomposition

A number of engineering problems have second-order ordinary differential equations as their mathematical models. In practice, we may have a large scale problem with a large number of degrees of freedom, which must be solved accurately. Therefore, treating the mathematical model governing the problems correctly is required in order to get an accurate solution. In this work, we use Adomian decomposition method to solve vibration models in the forms of initial value problems of second-order ordinary differential equations. However, for problems involving an external source, the Adomian decomposition method may not lead to an accurate solution if the external source is not correctly treated. In this paper, we propose a strategy to treat the external source when we implement the Adomian decomposition method to solve initial value problems of second-order ordinary differential equations. Computational results show that our strategy is indeed effective. We obtain accurate solutions to the considered problems. Note that exact solutions are often not available, so they need to be approximated using some methods, such as the Adomian decomposition method.

scholarly journals Comparison of the Solution of the Van der Pol Equation Using the Modified Adomian Decomposition Method and Truncated Taylor Series Method

2020 ◽  
pp. 105-113
Author(s):  
Joel Ndam ◽  
O. Adedire
Keyword(s):  
Taylor Series ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Analytic Method ◽  
Series Method ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Taylor Series Method

In this paper, we compare the solution of the van der Pol equation obtained by using the truncated Taylor series method and the modified Adomian decomposition method with the solution obtained by the Poincare-Lindstedt (P-L) method. The approximating 4-component modified Adomian decomposition method behaves more like an approximate P-L analytic method than the tenth-order Taylor series. Also, with the addition of just one term, the approximating 5-component modified Adomian decomposition method produces a more convergent solution to that of P-L analytic method than the twenty second-order Taylor series approximation as the independent variable t representing time progressively increases. A general comparison of the two solutions revealed that the absolute errors generated by the approximating polynomial from the Taylor series are greater than the ones generated from the modified Adomian decomposition method. It was further revealed that very few components of the modified Adomian decomposition could yield a series of about 3 times the order of the one obtained by using the Taylor series method. Hence, we recommend the inclusion of the modified Adomian Decomposition Method in modern mathematical tools.

scholarly journals Model of the telegraph line and its numerical solution

2018 ◽  
Vol 8 (1) ◽  
pp. 10-17 ◽  
Author(s):  
Petr Veigend ◽  
Gabriela Nečasová ◽  
Václav Šátek
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Taylor Series ◽  
Variable Step Size ◽  
Time Interval ◽  
Variable Order ◽  
Series Method ◽  
Linear Solver ◽  
Taylor Series Method

Abstract This paper deals with a model of the telegraph line that consists of system of ordinary differential equations, rather than partial differential telegraph equation. Numerical solution is then based on an original mathematical method. This method uses the Taylor series for solving ordinary differential equations with initial condition - initial value problems in a non-traditional way. Systems of ordinary differential equations are solved using variable order, variable step-size Modern Taylor Series Method. The Modern Taylor Series Method is based on a recurrent calculation of the Taylor series terms for each time interval. The second part of paper presents the solution of linear problems which comes from the model of telegraph line. All experiments were performed using MATLAB software, the newly developed linear solver that uses Modern Taylor Series Method. Linear solver was compared with the state of the art solvers in MATLAB and SPICE software.

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