scholarly journals Certain expansion formulae of incomplete $I$-functions associated with the Leibniz rule

2021 ◽  
Vol 2 (1) ◽  
pp. 42-50
Author(s):  
Sapna Meena ◽  
Sanjay Bhatter ◽  
Kamlesh Jangid ◽  
Sunil Dutt Purohit
Keyword(s):  
Leibniz Rule ◽  
Liouville Type ◽  
Expansion Formulae

In this paper, we determine some expansion formulae of the incomplete I-functions in affiliation with the Leibniz rule for the Riemann-Liouville type derivatives. Further, expansion formulae of the incomplete $\overline{I}$-function, incomplete $\overline{H}$-function, and incomplete H-function are conferred as extraordinary instances of our primary outcomes.

DERIVATIVE INVERSE SERIES RELATIONS AND LAGRANGE EXPANSION FORMULA

2013 ◽  
Vol 09 (04) ◽  
pp. 1001-1013 ◽  
Author(s):  
WENCHANG CHU
Keyword(s):  
Analytic Functions ◽  
Leibniz Rule ◽  
Expansion Formula ◽  
Connection Coefficients ◽  
Expansion Formulae ◽  
Higher Derivatives ◽  
Lagrange Expansion ◽  
Derivatives Of

A new pair of inverse series relations is established with the connection coefficients being expressed as higher derivatives of two fixed analytic functions. As applications, new proofs for the Lagrange expansion formulae are presented and Pfaff–Cauchy's generalizations of the Leibniz rule are reviewed.

BLOW-UP ANALYSIS FOR LIOUVILLE TYPE EQUATION IN SELF-DUAL GAUGE FIELD THEORIES

2005 ◽  
Vol 07 (02) ◽  
pp. 177-205 ◽  
Author(s):  
HIROSHI OHTSUKA ◽  
TAKASHI SUZUKI
Keyword(s):  
Gauge Field ◽  
Blow Up ◽  
Type Equation ◽  
Singular Part ◽  
Gauge Field Theories ◽  
Field Theories ◽  
Liouville Type ◽  
Blow Up Analysis ◽  
The Green Function ◽  
Dirac Measures

We study the asymptotic behavior of the solution sequence of Liouville type equations observed in various self-dual gauge field theories. First, we show that such a sequence converges to a measure with a singular part that consists of Dirac measures if it is not compact in W1,2. Then, under an additional condition, the singular limit is specified by the method of symmetrization of the Green function.

scholarly journals On a Riemann–Liouville Type Implicit Coupled System via Generalized Boundary Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1205
Author(s):  
Usman Riaz ◽  
Akbar Zada ◽  
Zeeshan Ali ◽  
Ioan-Lucian Popa ◽  
Shahram Rezapour ◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Coupled System ◽  
Initial Boundary ◽  
Liouville Type ◽  
Liouville Derivative ◽  
Banach Contraction ◽  
Generalized Boundary Conditions

We study a coupled system of implicit differential equations with fractional-order differential boundary conditions and the Riemann–Liouville derivative. The existence, uniqueness, and at least one solution are established by applying the Banach contraction and Leray–Schauder fixed point theorem. Furthermore, Hyers–Ulam type stabilities are discussed. An example is presented to illustrate our main result. The suggested system is the generalization of fourth-order ordinary differential equations with anti-periodic, classical, and initial boundary conditions.

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