120 research outputs found
Numerical resolution of Emden's equation using Adomian polynomials
Purpose: In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that appears in the study of the thermal behaviour of a spherical cloud and of the gravitational potential of a polytropic fluid at hydrostatic equilibrium. Design/methodology/approach: In their work, the authors first review Emden's equation and its possible solutions using the Frobenius and power series methods; then, Adomian polynomials are introduced. Afterwards, Emden's equation is solved using Adomian's decomposition method and, finally, they conclude with a comparison of the solution given by Adomian's method with the solution obtained by the other methods, for certain cases where the exact solution is known. Findings: Solving Emden's equation for n in the interval [0, 5] is very interesting for several scientific applications, such as astronomy. However, the exact solution is known only for n=0, n=1 and n=5. The experiments show that Adomian's method achieves an approximate solution which overlaps with the exact solution when n=0, and that coincides with the Taylor expansion of the exact solutions for n=1 and n=5. As a result, the authors obtained quite satisfactory results from their proposal. Originality/value: The main classical methods for obtaining approximate solutions of Emden's equation have serious computational drawbacks. The authors make a new, efficient numerical implementation for solving this equation, constructing iteratively the Adomian polynomials, which leads to a solution of Emden's equation that extends the range of variation of parameter n compared to the solutions given by both the Frobenius and the power series methods.This work has been supported by the Ministerio de Ciencia e Innovación, project TIN2009-10581
An efficient computational technique for solving the Fokker–Planck equation with space and time fractional derivatives
AbstractThis paper presents numerical solutions of the linear and nonlinear Fokker–Planck partial differential equations [FPPDEs] with space and time fractional derivatives through analytical solutions. These are treated by two analytical methods, namely, fractional reduced differential transform method [FRDTM] and fractional variational iteration method [FVIM] followed by some examples. Numerical results obtained by both FRDTM and FVIM are compared with some existing methods in the literature. This comparison shows the supremacy of FRDTM over FVIM and existing methods in terms of accuracy, simplicity and reliability
Hall Current and Joule Heating Effects on Flow of Couple Stress Fluid with Entropy Generation
In this work, an analytical study of the effects of Hall
current and Joule heating on the entropy generation rate of
couple stress fluid is performed. It is assumed that the applied
pressure gradient induces fluid motion. At constant velocity, hot
fluid is injected at the lower wall and sucked off at the upper
wall. The obtained equations governing the flow are transformed
to dimensionless form and the resulting nonlinear coupled
boundary value problems for velocity and temperature profiles
are solved by Adomian decomposition method. Analytical
expressions for fluid velocity and temperature are used to obtain
the entropy generation and the irreversibility ratio. The effects of
Hall current, Joule heating, suction/injection and magnetic field
parameters are presented and discussed through graphs. It is
found that Hall current enhances both primary and secondary
velocities and entropy generation. It is also interesting that Joule
heating raises fluid temperature and encourages entropy
production. On the other hand Hartman number inhibits fluid
motion while increase in suction/injection parameter leads to a
shift in flow symmetry
Analytical Solutions of the One-Dimensional Heat Equations Arising in Fractal Transient Conduction with Local Fractional Derivative
The one-dimensional heat equations with the heat generation arising in fractal transient conduction associated with local fractional derivative operators are investigated. Analytical solutions are obtained by using the local fractional Adomian decomposition method via local fractional calculus theory. The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique
Towards Faster-than-real-time Power System Simulation Using a Semi-analytical Approach and High-performance Computing
This dissertation investigates two possible directions of achieving faster-than-real-time simulation of power systems. The first direction is to develop a semi-analytical solution which represents the nonlinear dynamic characteristics of power systems in a limited time period. The second direction is to develop a parallel simulation scheme which allows the local numerical solutions of power systems to be developed independently in consecutive time intervals and then iteratively corrected toward the accurate global solution through the entire simulation time period.
For the first direction, the semi-analytical solution is acquired using Adomian decomposition method (ADM). The ADM assumes the analytical solution of any nonlinear system can be decomposed into the summation of infinite analytical expressions. Those expressions are derived recursively using the system differential equations. By only keeping a finite number of those analytical expressions, an approximation of the analytical solution is yielded, which is defined as a semi-analytical solution. The semi-analytical solutions can be developed offline and evaluated online to facilitate the speedup of simulations. A parallel implementation and variable time window approach for the online evaluation stage are proposed in addition to the time performance analysis.
For the second direction, the Parareal-in-time algorithm is tested for power system simulation. Parareal is essentially a multiple shooting method. It decomposes the simulation time into coarse time intervals and then fine time intervals within each coarse interval. The numerical integration uses a computational cheap solver on the coarse time grid and an expensive solver on the fine time grids. The solution within each coarse interval is propagated independently using the fine solver. The mismatch of the solution between the coarse solution and fine solution is corrected iteratively. The theoretical speedup can be achieved is the ratio of the coarse interval number and iteration number. In this dissertation, the Parareal algorithm is tested on the North American eastern interconnection system with around 70,000 buses and 5,000 generators
Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation
Fractional calculus is related to derivatives and integrals with the order is not an integer. Fractional Black-Scholes partial differential equation to determine the price of European-type call options is an application of fractional calculus in the economic and financial fields. Laplace decomposition method is one of the reliable and effective numerical methods for solving fractional differential equations. Thus, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes equation, where the fractional derivative used is the Caputo sense. Two numerical illustrations are presented in this paper. The results show that the Laplace decomposition method is an efficient, easy and very useful method for finding solutions of fractional Black-Scholes partial differential equations and boundary conditions for European option pricing problems
A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions
In this paper, the fractional order of rational Bessel functions collocation
method (FRBC) to solve Thomas-Fermi equation which is defined in the
semi-infinite domain and has singularity at and its boundary condition
occurs at infinity, have been introduced. We solve the problem on semi-infinite
domain without any domain truncation or transformation of the domain of the
problem to a finite domain. This approach at first, obtains a sequence of
linear differential equations by using the quasilinearization method (QLM),
then at each iteration solves it by FRBC method. To illustrate the reliability
of this work, we compare the numerical results of the present method with some
well-known results in other to show that the new method is accurate, efficient
and applicable
Solving Fokker-Planck Equations on Cantor Sets Using Local Fractional Decomposition Method
The local fractional decomposition method is applied to approximate the solutions for Fokker-Planck equations on Cantor sets with local fractional derivative. The obtained results give the present method that is very effective and simple for solving the differential equations on Cantor set
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