scholarly journals Kummer’s 24 Solutions of the Hypergeometric Differential Equation with the Aid of Fractional Calculus

2016 ◽  
Vol 06 (03) ◽  
pp. 180-191 ◽  
Author(s):  
Tohru Morita ◽  
Ken-ichi Sato
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Hypergeometric Differential Equation

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

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 262 ◽  
Author(s):  
Shengfeng Li ◽  
Yi Dong
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Hypergeometric Series ◽  
Second Order ◽  
Series Solutions ◽  
Frobenius Method ◽  
General Solutions ◽  
Hypergeometric Differential Equation ◽  
Polynomial Term ◽  
Hypergeometric Equations

In this paper, we expound on the hypergeometric series solutions for the second-order non-homogeneous k-hypergeometric differential equation with the polynomial term. The general solutions of this equation are obtained in the form of k-hypergeometric series based on the Frobenius method. Lastly, we employ the result of the theorem to find the solutions of several non-homogeneous k-hypergeometric differential equations.

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

scholarly journals THE GENERALIZED MATRIX VALUED HYPERGEOMETRIC EQUATION

2010 ◽  
Vol 21 (02) ◽  
pp. 145-155 ◽  
Author(s):  
P. ROMÁN ◽  
S. SIMONDI
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Hypergeometric Functions ◽  
Spherical Functions ◽  
Hypergeometric Equation ◽  
Necessary Condition ◽  
Hypergeometric Differential Equation ◽  
The Matrix ◽  
Generalized Matrix

The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in [4]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao's equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in [4] to identify the corresponding matrix hypergeometric functions nFm. We prove that, if n = m + 1, these functions are analytic for |z| < 1 and we give a necessary condition for the convergence on the unit circle |z| = 1.

Hypergeometric Differential Equation

1991 ◽  
pp. 25-116
Author(s):  
Katsunori Iwasaki ◽  
Hironobu Kimura ◽  
Shun Shimomura ◽  
Masaaki Yoshida
Keyword(s):  
Differential Equation ◽  
Hypergeometric Differential Equation

scholarly journals Affine Schwarz Map for the Hypergeometric Differential Equation

2008 ◽  
Vol 51 (2) ◽  
pp. 281-305 ◽  
Author(s):  
Ryoichi Kobayashi ◽  
Tatsuya Nishizaka ◽  
Shoji Shinzato ◽  
Masaaki Yoshida
Keyword(s):  
Differential Equation ◽  
Hypergeometric Differential Equation

scholarly journals Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales

2016 ◽  
Vol 2016 ◽  
pp. 1-21 ◽  
Author(s):  
Yanning Wang ◽  
Jianwen Zhou ◽  
Yongkun Li
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Fractional Differential Equation ◽  
Time Scales ◽  
Fractional Derivatives ◽  
Point Theory ◽  
Existence Results ◽  
Recent Approach ◽  
Equation Boundary ◽  
Fractional Differential

Using conformable fractional calculus on time scales, we first introduce fractional Sobolev spaces on time scales, characterize them, and define weak conformable fractional derivatives. Second, we prove the equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, uniform convexity, and compactness of some imbeddings, which can be regarded as a novelty item. Then, as an application, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for ap-Laplacian conformable fractional differential equation boundary value problem on time scaleT:  Tα(Tαup-2Tα(u))(t)=∇F(σ(t),u(σ(t))),Δ-a.e.  t∈a,bTκ2,u(a)-u(b)=0,Tα(u)(a)-Tα(u)(b)=0,whereTα(u)(t)denotes the conformable fractional derivative ofuof orderαatt,σis the forward jump operator,a,b∈T,  0<a<b,  p>1, andF:[0,T]T×RN→R. By establishing a proper variational setting, we obtain three existence results. Finally, we present two examples to illustrate the feasibility and effectiveness of the existence results.

Sign in / Sign up

Export Citation Format

Share Document