scholarly journals Estimation of divergences on time scales via the Green function and Fink’s identity

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Iqrar Ansari ◽  
Khuram Ali Khan ◽  
Ammara Nosheen ◽  
Ðilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Green Function ◽  
Time Scales ◽  
Convex Functions ◽  
Quantum Calculus ◽  
Divergence Measures ◽  
The Green Function ◽  
Fink’S Identity

AbstractThe aim of the present paper is to obtain new generalizations of an inequality for n-convex functions involving Csiszár divergence on time scales using the Green function along with Fink’s identity. Some new results in h-discrete calculus and quantum calculus are also presented. Moreover, inequalities for some divergence measures are also deduced.

scholarly journals Estimation of divergence measures on time scales via Taylor’s polynomial and Green’s function with applications in q-calculus

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Iqrar Ansari ◽  
Khuram Ali Khan ◽  
Ammara Nosheen ◽  
Ðilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Green’S Function ◽  
Time Scales ◽  
Green's Function ◽  
Convex Functions ◽  
Higher Order ◽  
Quantum Calculus ◽  
Divergence Measures ◽  
New Inequalities

AbstractTaylor’s polynomial and Green’s function are used to obtain new generalizations of an inequality for higher order convex functions containing Csiszár divergence on time scales. Various new inequalities for some divergence measures in quantum calculus and h-discrete calculus are also established.

scholarly journals Some inequalities for Csiszár divergence via theory of time scales

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Iqrar Ansari ◽  
Khuram Ali Khan ◽  
Ammara Nosheen ◽  
Ðilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Probability Measures ◽  
Quantum Calculus ◽  
Divergence Measures

AbstractIn this paper, we present some inequalities for Csiszár f-divergence between two probability measures on time scale. These results extend some known results in the literature and offer new results in h-discrete calculus and quantum calculus. We also present several inequalities for divergence measures.

scholarly journals Estimation of divergence measures via weighted Jensen inequality on time scales

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Iqrar Ansari ◽  
Khuram Ali Khan ◽  
Ammara Nosheen ◽  
Ðilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Time Scales ◽  
Lower Bounds ◽  
Time Scale ◽  
Jensen’S Inequality ◽  
Jensen Inequality ◽  
Jensen's Inequality ◽  
Quantum Calculus ◽  
Divergence Measures ◽  
Zipf Law ◽  
New Inequalities

AbstractThe main purpose of the presented paper is to obtain some time scale inequalities for different divergences and distances by using weighted time scales Jensen’s inequality. These results offer new inequalities in h-discrete calculus and quantum calculus and extend some known results in the literature. The lower bounds of some divergence measures are also presented. Moreover, the obtained discrete results are given in the light of the Zipf–Mandelbrot law and the Zipf law.

scholarly journals Positivity of sums and integrals for n-convex functions via the Fink identity and new Green functions

2021 ◽  
Vol 66 (4) ◽  
pp. 613-627
Author(s):  
Asif R. Khan ◽  
◽  
Josip Pecaric ◽  
Keyword(s):  
Green Function ◽  
Convex Functions ◽  
Mean Value ◽  
Higher Order ◽  
Green Functions ◽  
Mean Value Theorems ◽  
The Green Function

We consider positivity of sum $\sum_{i=1}^np_if(x_i)$ involving convex functions of higher order. Analogous for integral $\int_a^bp(x)f(g(x))dx$ is also given. Representation of a function $f$ via the Fink identity and the Green function leads us to identities for which we obtain conditions for positivity of the mentioned sum and integral. We obtain bounds for integral remainders which occur in those identities as well as corresponding mean value theorems.

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

New generalized 2D Ostrowski type inequalities on time scales with k2 points using a parameter

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3155-3169 ◽  
Author(s):  
Seth Kermausuor ◽  
Eze Nwaeze
Keyword(s):  
Numerical Analysis ◽  
Time Scales ◽  
Type Inequality ◽  
Dimensional Case ◽  
Multiple Points ◽  
Quantum Calculus ◽  
Independent Variables ◽  
Ostrowski Type Inequality ◽  
Two Independent Variables

Recently, a new Ostrowski type inequality on time scales for k points was proved in [G. Xu, Z. B. Fang: A Generalization of Ostrowski type inequality on time scales with k points. Journal of Mathematical Inequalities (2017), 11(1):41-48]. In this article, we extend this result to the 2-dimensional case. Besides extension, our results also generalize the three main results of Meng and Feng in the paper [Generalized Ostrowski type inequalities for multiple points on time scales involving functions of two independent variables. Journal of Inequalities and Applications (2012), 2012:74]. In addition, we apply some of our theorems to the continuous, discrete, and quantum calculus to obtain more interesting results in this direction. We hope that results obtained in this paper would find their place in approximation and numerical analysis.

The Green function method for the two-surface problem

1970 ◽  
Vol 8 (13) ◽  
pp. 1069-1071 ◽  
Author(s):  
F. Flores ◽  
F. Garcia-Moliner ◽  
J. Rubio
Keyword(s):  
Green Function ◽  
Function Method ◽  
Green Function Method ◽  
The Green Function

Electrostatic oscillations in cold inhomogeneous plasma I. Differential equation approach

1971 ◽  
Vol 5 (2) ◽  
pp. 239-263 ◽  
Author(s):  
Z. Sedláček
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Normal Mode Analysis ◽  
Inhomogeneous Plasma ◽  
Mode Analysis ◽  
Complex Frequency ◽  
Plasma Oscillations ◽  
The Laplace Transform ◽  
Collective Oscillations ◽  
The Green Function

Small amplitude electrostatic oscillations in a cold plasma with continuously varying density have been investigated. The problem is the same as that treated by Barston (1964) but instead of his normal-mode analysis we employ the Laplace transform approach to solve the corresponding initial-value problem. We construct the Green function of the differential equation of the problem to show that there are branch-point singularities on the real axis of the complex frequency-plane, which correspond to the singularities of the Barston eigenmodes and which, asymptotically, give rise to non-collective oscillations with position-dependent frequency and damping proportional to negative powers of time. In addition we find an infinity of new singularities (simple poles) of the analytic continuation of the Green function into the lower half of the complex frequency-plane whose position is independent of the spatial co-ordinate so that they represent collective, exponentially damped modes of plasma oscillations. Thus, although there may be no discrete spectrum, in a more general sense a dispersion relation does exist but must be interpreted in the same way as in the case of Landau damping of hot plasma oscillations.

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