The Grünwald–Letnikov method for fractional differential equations

AbstractThis paper is devoted to the numerical treatment of fractional differential equations. Based on the Grünwald–Letnikov definition of fractional derivatives, finite difference schemes for the approximation of the solution are discussed. The main properties of these explicit and implicit methods concerning the stability, the convergence and the error behavior are studied related to linear test equations. The asymptotic stability and the absolute stability of these methods are proved. Error representations and estimates for the truncation, propagation and global error are derived. Numerical experiments are given

Sobre operadores de integración fraccional y algoritmos computacionales

Algunos autores como Kalla, Saxena y Saigo, han definido y estudiado operadores de integración fraccional que tienen como núcleo la función hipergeométrica de Gauss. El propósito de este trabajo es estudiar tales operadores con la función R-∝(b1,b2;t,X) de Carlson como núcleo, así como también presentar algoritmos para el cálculo de semidiferintegrales.Kalla, Saxena, Saigo and some other authors, have defined and studied fractional operators with the Gauss hypergeometric function as nucleus. The object of this paper is to study operators containing the Carlson's function R-∝(b1,b2;t,X); as well as to present and discuss algorithms for numerical calculations of semidiferintegrals

Use of generalized hypergeometric functions in analytic stellar models

The present paper deals with the technique of integration theory of special functions applied to two simple analytic stellar models. We consider two cases, one with a non-linear dependence of the radial density and the other with a generalized energy generation rate. The integration theory of the generalized hypergeometric functions is applied to evaluate analitically the rate of nuclear energy generation. Some known results follow as particular cases of our formulae established here

On a class of generalized analytic functions

This paper deals with a new generalization of analytical functions.The p-wave functions are introduced and studied. We consider their theoretical aspect and applications. Some integral representations of x^k y^l -wave functions (k, l − const. > 0), and their inversion formulas are derived. As an application of the theory, a singular Cauchy problem is formulated and solved in terms of the Bessel function of the first kind and Gauss hypergeometric function

On Grüss type inequality for a hypergeometric fractional integral

Aim of the present paper is to investigate a new integral inequality of Grüss type for a hypergeometric fractional integral. Two main results are proved, the first one deals with Grüss type inequality using the hypergeometric fractional integral. The second result states another inequality regarding two synchronous functions

A generalization of elliptic-type integrals

The method o f steepest descent i s employed t o o b t a i n r e l a t i o n s between KV(k,m), sU(k,v) and incornplete gamma f u n c t i o n s . W e t a b u l a t e these e l -1 t p t i c -t y p e i n t e g r a l s by u s i n g s u i t a b l e formulae. Some k n o w n r e s u l t s f o l l o w as p a r t i c u l a r cases o f o u r formulae e s t a b l i s h e d here. I n a r e c e n t paperl, t h e a u t h o r s have s t u d i e d a fami l y o f i n t eg r a l s where O $ k < 1, ~e ( u ) > -and m i s a non-negative i n t e g e r . For m = O and = j , a p o s i t i v e i n t e g e r where O < k < 1, a fami 1 y o f i n t e g r a l s considered by E p s t e i n and Hubbel 12.such i n t e g r a l s a r e found i n t h e a p p l i c a t i o n o f t h e Legendre polynomial expans i o n method 3 t o c e r t a i n problems i n v o l v i n g computation o f t h e r a d i a t i o

Some results on generalized elliptic-type integrals

Abstract In this paper we study a fam r TI ly of integra Is of the form where O 6 k < 1, Re(p) > -and m is a nonnegative integer. Such integrals occur in radiation field problems. We obtain a series expansion and establish its relationship with Gauss' hypergeometric function. Asymptotlc expansions val id in the neighbourhood of k2=l are given.0ne of these formulas has been obtained by the use of an Abel ian theorem. Some recurrence relations are established. Results obtained earlier by Epstein and Hubbell, and Weiss follow as particular cases of our formulae given here. Some numerical values of ~~ ( k , m ) for selected values of the parameter are tabulated, using different formulae

On a new approach to convolution constructions

In this paper we establish some new approach to constructing convolution for general Mellin type transforms. This method is based on the theory of double Hellin-Barnes integrals. Some properties of convolutions and several examples are given

On the Lauwerier formulation of the temperature field problem in oil strata

The paper is concerned with the fractional extension of the Lauwerier formulation of the problem related to the temperature field description in a porous medium (sandstone) saturated with oil (strata). The boundary value problem for the fractional heat equation is solved by means of the Caputo differintegration operator D∗(α) of order 0<α≤1 and the Laplace transform. The solution is obtained in an integral form, where the integrand is expressed in terms of a convolution of two special functions of Wright type

Generalized Differential Transform Method to Space-Time Fractional Telegraph Equation

We use generalized differential transform method (GDTM) to derive the solution of space-time fractional telegraph equation in closed form. The space and time fractional derivatives are considered in Caputo sense and the solution is obtained in terms of Mittag-Leffler functions