897 research outputs found
An approximate solution of the MHD Falkner-Skan flow by Hermite functions pseudospectral method
Based on a new approximation method, namely pseudospectral method, a solution
for the three order nonlinear ordinary differential laminar boundary layer
Falkner-Skan equation has been obtained on the semi-infinite domain. The
proposed approach is equipped by the orthogonal Hermite functions that have
perfect properties to achieve this goal. This method solves the problem on the
semi-infinite domain without truncating it to a finite domain and transforming
domain of the problem to a finite domain. In addition, this method reduces
solution of the problem to solution of a system of algebraic equations. We also
present the comparison of this work with numerical results and show that the
present method is applicable.Comment: 15 pages, 4 figures; Published online in the journal of
"Communications in Nonlinear Science and Numerical Simulation
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Many PDEs involving fractional Laplacian are naturally set in unbounded
domains with underlying solutions decay very slowly, subject to certain power
laws. Their numerical solutions are under-explored. This paper aims at
developing accurate spectral methods using rational basis (or modified mapped
Gegenbauer functions) for such models in unbounded domains. The main building
block of the spectral algorithms is the explicit representations for the
Fourier transform and fractional Laplacian of the rational basis, derived from
some useful integral identites related to modified Bessel functions. With these
at our disposal, we can construct rational spectral-Galerkin and direct
collocation schemes by pre-computing the associated fractional differentiation
matrices. We obtain optimal error estimates of rational spectral approximation
in the fractional Sobolev spaces, and analyze the optimal convergence of the
proposed Galerkin scheme. We also provide ample numerical results to show that
the rational method outperforms the Hermite function 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
Numerical approximations for population growth model by Rational Chebyshev and Hermite Functions collocation approach: A comparison
This paper aims to compare rational Chebyshev (RC) and Hermite functions (HF)
collocation approach to solve the Volterra's model for population growth of a
species within a closed system. This model is a nonlinear integro-differential
equation where the integral term represents the effect of toxin. This approach
is based on orthogonal functions which will be defined. The collocation method
reduces the solution of this problem to the solution of a system of algebraic
equations. We also compare these methods with some other numerical results and
show that the present approach is applicable for solving nonlinear
integro-differential equations.Comment: 18 pages, 5 figures; Published online in the journal of "Mathematical
Methods in the Applied Sciences
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