294 research outputs found
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
Numerical solution of variable-order time fractional weakly singular partial integro-differential equations with error estimation
In this paper, we apply Legendre-Laguerre functions (LLFs) and collocation method to obtain the approximate solution of variable-order time-fractional partial integro-differential equations (VO-TF-PIDEs) with the weakly singular kernel. For this purpose, we derive the pseudo-operational matrices with the use of the transformation matrix. The collocation method and pseudo-operational matrices transfer the problem to a system of algebraic equations. Also, the error analysis of the proposed method is given. We consider several examples to illustrate the proposed method is accurate
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
a collocation method via the quasi affine biorthogonal systems for solving weakly singular type of volterra fredholm integral equations
Abstract Tight framelet system is a recently developed tool in applied mathematics. Framelets, due to their nature, are widely used in the area of image manipulation, data compression, numerical analysis, engineering mathematical problems such as inverse problems, visco-elasticity or creep problems, and many more. In this manuscript we provide a numerical solution of important weakly singular type of Volterra - Fredholm integral equations WSVFIEs using the collocation type quasi-affine biorthogonal method. We present a new computational method based on special B-spline tight framelets and use it to introduce our numerical scheme. The method provides a robust solution for the given WSVFIE by using the resulting matrices based on these biorthogonal wavelet. We demonstrate the validity and accuracy of the proposed method by some numerical examples
A new numerical technique based on Chelyshkov polynomials for solving two-dimensional stochastic It\^o-Volterra Fredholm integral equation
In this paper, a two-dimensional operational matrix method based on
Chelyshkov polynomials is implemented to numerically solve the two-dimensional
stochastic It\^o-Volterra Fredholm integral equations. These equations arise in
several problems such as an exponential population growth model with several
independent white noise sources. In this paper a new stochastic operational
matrix has been derived first time ever by using Chelyshkov polynomials. After
that, the operational matrices are used to transform the It\^o-Volterra
Fredholm integral equation into a system of linear algebraic equations by using
Newton cotes nodes as collocation point that can be easily solved. Furthermore,
the convergence and error bound of the suggested method are well established.
In order to illustrate the effectiveness, plausibility, reliability, and
applicability of the existing technique, two typical examples have been
presented
Numerical solution of fractional Fredholm integro-differential equations by spectral method with fractional basis functions
This paper presents an efficient spectral method for solving the fractional
Fredholm integro-differential equations. The non-smoothness of the solutions to
such problems leads to the performance of spectral methods based on the
classical polynomials such as Chebyshev, Legendre, Laguerre, etc, with a low
order of convergence. For this reason, the development of classic numerical
methods to solve such problems becomes a challenging issue. Since the
non-smooth solutions have the same asymptotic behavior with polynomials of
fractional powers, therefore, fractional basis functions are the best candidate
to overcome the drawbacks of the accuracy of the spectral methods. On the other
hand, the fractional integration of the fractional polynomials functions is in
the class of fractional polynomials and this is one of the main advantages of
using the fractional basis functions. In this paper, an implicit spectral
collocation method based on the fractional Chelyshkov basis functions is
introduced. The framework of the method is to reduce the problem into a
nonlinear system of equations utilizing the spectral collocation method along
with the fractional operational integration matrix. The obtained algebraic
system is solved using Newton's iterative method. Convergence analysis of the
method is studied. The numerical examples show the efficiency of the method on
the problems with smooth and non-smooth solutions in comparison with other
existing methods
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