scholarly journals Efficient multiple-term approximations for the generalised elliptic-type integrals

1997 ◽  
Vol 38 (3) ◽  
pp. 411-426 ◽  
Author(s):  
M. El-Gabali
Keyword(s):  
Entire Range ◽  
Special Functions ◽  
Gauss Hypergeometric Function ◽  
Elliptic Type ◽  
Closed Form Solutions ◽  
Novel Technique ◽  
Single Term ◽  
Type Integral ◽  
Image Position ◽  
Term Approximation

AbstractThe generalised elliptic-type integral Rμ(k, α, γ)where 0 ≤ k < 1, Re(γ) > Re(α) > 0, Re(μ) ≥ −0.5, is represented in terms of the Gauss hypergeometric function by Kalla, Conde and Hubbell [8]. In 1987, Kalla, Lubner and Hubbell derived a simple-structured single-term approximation for this function in the neighbourhood of k2 = 1 in some range of the parameters α, γ and μ. Another formula which complements the parameter range was recently derived by the author. In this paper a novel technique is used in deriving multiple-term efficient approximations (in the neighbourhood of k2 = 1) which may be considered as a generalisation to the concept of the single-term approximations mentioned above. Two non-overlapping expressions which almost cover the entire range of parameters (α, γ, μ) are derived. Closed-form solutions are obtained for single- and double-term approximations (in the neighbourhood of k2 = 1). Results show that the proposed technique is superior to existing approximations for the same number of terms. Our formulation has potential application for a wide class of special functions.

scholarly journals Multiple-term approximations for Appell's F1 function

1997 ◽  
Vol 39 (1) ◽  
pp. 135-148
Author(s):  
Magdi A. El-Gabali
Keyword(s):  
Error Analysis ◽  
Closed Form ◽  
Wide Class ◽  
Special Functions ◽  
Form Error ◽  
Novel Technique ◽  
Highly Efficient ◽  
Image Position

AbstractIn this paper computational issues of Appell's F1 functionare addressed. A novel technique is used in the derivation of highly efficient multiple-term approximations of this function (including asymptotic ones). Simple structured single- and double-term approximations, as the first two candidates of this multiple-term representa-tion, are developed in closed form. Error analysis shows that- the developed algorithms are superior to existing approximations for the same number of terms. The formulation presented is highly efficient and could be extended to a wide class of special functions.

scholarly journals New fractional inequalities of Hermite–Hadamard type involving the incomplete gamma functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Dumitru Baleanu ◽  
Artion Kashuri ◽  
Faraidun Hamasalh ◽  
...  
Keyword(s):  
Convex Functions ◽  
Special Functions ◽  
Integral Inequalities ◽  
Fractional Operators ◽  
Gamma Functions ◽  
Incomplete Gamma Functions ◽  
Type Integral

AbstractA specific type of convex functions is discussed. By examining this, we investigate new Hermite–Hadamard type integral inequalities for the Riemann–Liouville fractional operators involving the generalized incomplete gamma functions. Finally, we expose some examples of special functions to support the usefulness and effectiveness of our results.

scholarly journals Study of Incomplete Elliptic Integrals Pertaining to pΨq Function

2018 ◽  
Vol 14 (2) ◽  
pp. 11-18 ◽  
Author(s):  
Ravi Shanker Dubey ◽  
Anil Sharma ◽  
Monika Jain
Keyword(s):  
Fracture Mechanics ◽  
Integral Operator ◽  
Fractional Integral ◽  
Elliptic Integral ◽  
Elliptic Type ◽  
Liouville Type ◽  
Definite Integrals ◽  
Type Integral ◽  
Geometric Representations ◽  
Incomplete Elliptic Integrals

Abstract Elliptic-type integral plays a major role in the study of different problems of physics and technology including fracture mechanics. Many papers have been written for various families of elliptic-type integrals. Due to their applications here, we are presenting an organized study of certain generalized family of incomplete elliptic integral. The obtained results are basic in nature have various generalizations. While using the fractional integral operator of Riemann-Liouville type, we found several obvious hyper geometric representations. Which are further used to originate many definite integrals relating to their modules and amplitude of elliptic type generalized incomplete integrals.

A Tensor Equation of Elliptic Type

1953 ◽  
Vol 5 ◽  
pp. 524-535 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Riemannian Manifold ◽  
Differential Equations ◽  
Tensor Field ◽  
Differential Operators ◽  
Elliptic Type ◽  
Tensor Equation ◽  
Tensor Theory ◽  
Elliptic Differential Equations ◽  
Invariant Differential ◽  
Image Position

The theory of the systems of partial differential equations which arise in connection with the invariant differential operators on a Riemannian manifold may be developed by methods based on those of potential theory. It is therefore natural to consider in the same context the theory of elliptic differential equations, in particular those which are self-adjoint. Some results for a tensor equation in which appears, in addition to the operator Δ of tensor theory, a matrix or double tensor field defined on the manifold, are here presented. The equation may be writtenin a notation explained below.

A Nonfield Analytical Method for Solving Energy Transport Equations

2020 ◽  
Vol 142 (4) ◽  
Author(s):  
Vladimir Kulish
Keyword(s):  
Analytical Method ◽  
Energy Transport ◽  
Differential Operators ◽  
Transport Equations ◽  
Theoretical Justification ◽  
Closed Form Solutions ◽  
The Past ◽  
Elegant Method ◽  
Type Integral ◽  
Transport Problems

Abstract In 2000, Kulish and Lage proposed an elegant method, which allows one to obtain analytical (closed-form) solutions to various energy transport problems. The solutions thus obtained are in the form of the Volterra-type integral equations, which relate the local values of an intensive property (e.g., temperature, mass concentration, and velocity) and the corresponding energy flux (e.g., heat flux, mass flux, and shear stress). The method does not require one to solve for the entire domain, and hence, is a nonfield analytical method. Over the past 19 years, the method was shown to be extremely effective when applied to solving numerous energy transport problems. In spite of all these developments, no general theoretical justification of the method was proposed until now. The present work proposes a justification of the procedure behind the method and provides a generalized technique of splitting the differential operators in the energy transport equations.

scholarly journals Definite Integral of Algebraic, Exponential and Hyperbolic Functions Expressed in Terms of Special Functions

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1425
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Function ◽  
Special Functions ◽  
Complex Numbers ◽  
Definite Integral ◽  
Hyperbolic Functions ◽  
Closed Form Solutions ◽  
Definite Integrals ◽  
Arbitrary Complex ◽  
Famous Book

While browsing through the famous book of Bierens de Haan, we came across a table with some very interesting integrals. These integrals also appeared in the book of Gradshteyn and Ryzhik. Derivation of these integrals are not listed in the current literature to best of our knowledge. The derivation of such integrals in the book of Gradshteyn and Ryzhik in terms of closed form solutions is pertinent. We evaluate several of these definite integrals of the form ∫0∞(a+y)k−(a−y)keby−1dy, ∫0∞(a+y)k−(a−y)keby+1dy, ∫0∞(a+y)k−(a−y)ksinh(by)dy and ∫0∞(a+y)k+(a−y)kcosh(by)dy in terms of a special function where k, a and b are arbitrary complex numbers.

On the Epstein-Hubbell generalized elliptic-type integral

1994 ◽  
Vol 61 (2-3) ◽  
pp. 157-161 ◽  
Author(s):  
Pavel G. Todorov ◽  
John H. Hubbell
Keyword(s):  
Elliptic Type ◽  
Type Integral

A Novel Analytical Implementation of Nonlinear Volterra Integral Equations

2012 ◽  
Vol 67 (12) ◽  
pp. 674-678 ◽  
Author(s):  
Majid Khan ◽  
Muhammad Asif Gondal ◽  
Syeda Iram Batool
Keyword(s):  
Exact Solutions ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Volterra Equations ◽  
Volterra Type ◽  
Transform Method ◽  
Novel Technique ◽  
Homotopy Perturbation ◽  
Type Integral ◽  
Nonlinear Volterra Equations

This article aims at preferring a new and viable algorithm, specifically a two-step homotopy perturbation transform algorithm (TSHPTA). This novel technique is a feasible way in finding exact solutions with a small amount of calculations. As a simple but typical example, it demonstrates the strength and the great potential of the two-step homotopy perturbation transform method to solve nonlinear Volterra-type integral equations efficiently. The results reveal that the proposed scheme is suitable for the nonlinear Volterra equations.

scholarly journals ON BRANNAN'S COEFFICIENT CONJECTURE AND APPLICATIONS

2007 ◽  
Vol 49 (1) ◽  
pp. 45-52 ◽  
Author(s):  
STEPHAN RUSCHEWEYH ◽  
LUIS SALINAS
Keyword(s):  
Hypergeometric Function ◽  
Unit Disk ◽  
Boundary Point ◽  
Univalent Functions ◽  
Gegenbauer Polynomials ◽  
Gauss Hypergeometric Function ◽  
Coefficient Estimates ◽  
Image Position

Abstract.D. Brannan's conjecture says that for 0 <α,β≤1, |x|=1, and n∈N one has |A2n−1(α,β,x)|≤|A2n−1(α,β,1)|, where We prove this for the case α=β, and also prove a differentiated version of the Brannan conjecture. This has applications to estimates for Gegenbauer polynomials and also to coefficient estimates for univalent functions in the unit disk that are ‘starlike with respect to a boundary point’. The latter application has previously been conjectured by H. Silverman and E. Silvia. The proofs make use of various properties of the Gauss hypergeometric function.

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