Particular Solutions of the Confluent Hypergeometric Differential Equation by Using the Nabla Fractional Calculus Operator

Abstract

Abstract:In this work; we present a method for solving the second-order linear ordinary differential equation of hypergeometric type. The solutions of this equation are given by the confluent hypergeometric functions (CHFs). Unlike previous studies, we obtain some different new solutions of the equation without using the CHFs. Therefore, we obtain new discrete fractional solutions of the homogeneous and non-homogeneous confluent hypergeometric differential equation (CHE) by using a discrete fractional Nabla calculus operator. Thus, we obtain four different new discrete complex fractional solutions for these equations.

k-Hypergeometric Series Solutions to One Type of Non-Homogeneous k-Hypergeometric Equations

“…Since then, Euler, too, had researched the topic on the hypergeometric series, but the first full systematic study was introduced by Gauss [12]. Some works and complete references concerning both the hypergeometric series and the certain equation 2can be found in Kummer [13], Riemann [14], Bailey [15,16], Chaundy [17], Srivastava [18], Whittaker [19], Beukers [20], Gasper [21], Olde Daalhuis [22,23], Dwork [24], Chu [25], Yilmazer et al [26], Morita et al [27], Abramov et al [28], Alfedeel et al [29], and the literature therein. However, in contrast to the extensive studies on Equation 2, other hypergeometric differential equations with k ∈ R + are very limited.…”

Discrete fractional solution of a nonhomogeneous non-Fuchsian differential equations

“…A similar theory was started for discrete fractional calculus and the definition and properties of fractional sums and differences theory were developed. Many article related to this topic have seemed lately [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Finding exact solutions to non-linear PDE defining the evolution of localized waveforms is an important subject in non-linear science [21][22][23].…”

On numerical solutions for the Caputo-Fabrizio fractional heat-like equation

“…[1][2][3][4]. Besides there has been a significant theoretical development in fractional differential equations and its applications [5][6][7][8][9][10]. On the other hand, fractional derivatives supply an important implement for the definition of hereditary characteristics of different necessaries and treatment.…”