122 research outputs found
Comparing numerical methods for the solutions of systems of ordinary differential equations
AbstractIn this article, we implement a relatively new numerical technique, the Adomian decomposition method, for solving linear and nonlinear systems of ordinary differential equations. The method in applied mathematics can be an effective procedure to obtain analytic and approximate solutions for different types of operator equations. In this scheme, the solution takes the form of a convergent power series with easily computable components. This paper will present a numerical comparison between the Adomian decomposition and a conventional method such as the fourth-order Runge-Kutta method for solving systems of ordinary differential equations. The numerical results demonstrate that the new method is quite accurate and readily implemented
Fractional Variational Iteration Method for Fractional Nonlinear Differential Equations
Recently, fractional differential equations have been investigated via the
famous variational iteration method. However, all the previous works avoid the
term of fractional derivative and handle them as a restricted variation. In
order to overcome such shortcomings, a fractional variational iteration method
is proposed. The Lagrange multipliers can be identified explicitly based on
fractional variational theory.Comment: 12 pages, 1 figure
On Two-Dimensional Fractional Chaotic Maps with Symmetries
In this paper, we propose two new two-dimensional chaotic maps with closed curve fixed points. The chaotic behavior of the two maps is analyzed by the 0–1 test, and explored numerically using Lyapunov exponents and bifurcation diagrams. It has been found that chaos exists in both fractional maps. In addition, result shows that the proposed fractional maps shows the property of coexisting attractors
Application of homotopy-perturbation method to fractional IVPs
Fractional initial-value problems (fIVPs) arise from many fields of physics and play a very important role in various branches of science and engineering. Finding accurate and efficient methods for solving fIVPs has become an active research undertaking. In this paper, both linear and nonlinear fIVPs are considered. Exact and/or approximate analytical solutions of the fIVPs are obtained by the analytic homotopy-perturbation method (HPM). The results of applying this procedure to the studied cases show the high accuracy, simplicity and efficiency of the approach
An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method
In this paper we propose a collocation method for solving some well-known
classes of Lane-Emden type equations which are nonlinear ordinary differential
equations on the semi-infinite domain. They are categorized as singular initial
value problems. The proposed approach is based on a Hermite function
collocation (HFC) method. To illustrate the reliability of the method, some
special cases of the equations are solved as test examples. The new method
reduces the solution of a problem to the solution of a system of algebraic
equations. Hermite functions have prefect properties that make them useful to
achieve this goal. We compare the present work with some well-known results and
show that the new method is efficient and applicable.Comment: 34 pages, 13 figures, Published in "Computer Physics Communications
Additional degrees of parallelism within the Adomian decomposition method
4th International Conference on Computational Engineering (ICCE 2017), 28-29 September 2017, DarmstadtThis is the author accepted manuscript. The final version is available from Springer via the DOI in this record.The trend of future massively parallel computer architectures challenges the exploration of additional degrees of parallelism also in the time dimension when solving continuum mechanical partial differential equations. The Adomian decomposition method (ADM) is investigated to this respects in the present work. This is accomplished by comparison with the Runge-Kutta (RK) time integration and put in the context of the viscous Burgers equation. Our studies show that both methods have similar restrictions regarding their maximal time step size. Increasing the order of the schemes leads to larger errors for the ADM compared to RK. However, we also discuss a parallelization within the ADM, reducing its runtime complexity from O(n^2) to O(n). This indicates the possibility to make it a viable competitor to RK, as fewer function evaluations have to be done in serial, if a high order method is desired. Additionally, creating ADM schemes of high-order is less complex as it is with RK.The work of Andreas Schmitt is supported by the ’Excellence
Initiative’ of the German Federal and State Governments and the Graduate
School of Computational Engineering at Technische Universit¨at Darmstadt
APPLICATION OF REPRODUCING KERNEL ALGORITHM FOR SOLVING DIRICHLET TIME-FRACTIONAL DIFFUSION-GORDON TYPES EQUATIONS IN POROUS MEDIA
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