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Papers by Tahir ullah khan

$Research paper thumbnail of A new generalized Hilfer-type fractional derivative with applications to space-time diffusion equation$

Results in Physics

Abstract This paper is concerned to present and apply a new generalized fractional derivative, th... more Abstract This paper is concerned to present and apply a new generalized fractional derivative, that is the Generalized Hilfer-type (GH) fractional derivative. This derivative unifies various previously defined fractional derivatives of the types Hilfer-Katugampola, Hilfer-Hadamard, Caputo-Hadamard, Hadamard, Hilfer, Riemann-Liouville, Caputo etc into a single form. The mellin transform of this new GH fractional derivative is obtained. As an application, a generalized space-time fractional diffusion equation is constructed using the newly obtained GH fractional derivative in the time-variable and the already existing Riesz fractional derivative in the space-variable. For the solution of this new generalized fractional diffusion equation, the similarity transformations and the mellin transform methods are used where an explicit solution in terms of the Fox’s H-function is derived. The diffusive behavior and the role of various parameters involved in the newly obtained GH fractional derivative are described with the help of 3D plots.

Journal of Function Spaces, 2022

The main objective of the paper is to develop an innovative idea of bringing continuous and discr... more The main objective of the paper is to develop an innovative idea of bringing continuous and discrete inequalities into a unified form. The desired objective is thus obtained by embedding majorization theory with the existing notion of continuous inequalities. These notions are applied to the latest generalized form of the inequalities, popularly known as the Hermite–Hadamard–Jensen–Mercer inequalities. Moreover, the frequently-used Caputo fractional operators are employed, which are rightly considered critical, especially for applied problems. Both weighted and unweighted forms of the developed results are discussed. In addition to this, some bounds are also provided for the absolute difference between the left- and right-sides of the main results.

$Research paper thumbnail of Hermite-Hadamard inequality for new generalized conformable fractional operators$

AIMS Mathematics, 2021

This paper is concerned to establish an advanced form of the well-known Hermite-Hadamard (HH) ine... more This paper is concerned to establish an advanced form of the well-known Hermite-Hadamard (HH) inequality for recently-defined Generalized Conformable (GC) fractional operators. This form of the HH inequality combines various versions (new and old) of this inequality, containing operators of the types Katugampula, Hadamard, Riemann-Liouville, conformable and Riemann, into a single form. Moreover, a novel identity containing the new GC fractional integral operators is proved. By using this identity, a bound for the absolute of the difference between the two rightmost terms in the newly-established Hermite-Hadamard inequality is obtained. Also, some relations of our results with the already existing results are presented. Conclusion and future works are presented in the last section.

Fasciculi Mathematici, 2017

In this article first, we give an integral identity and prove some Hermite-Hadamard type inequali... more In this article first, we give an integral identity and prove some Hermite-Hadamard type inequalities for the function f such that |f″|q is convex or concave for q ≥ 1. Second, by using these results, we present applications to f-divergence measures. At the end, we obtain some bounds for special means of real numbers and new error estimates for the trapezoidal formula.

$Research paper thumbnail of Generalized conformable fractional operators$

Journal of Computational and Applied Mathematics, 2019

In this paper we are concerned with the problem u (α) (t) = Au(t) + f (t, u(t)) t ∈ [0, T ] u(0) ... more In this paper we are concerned with the problem u (α) (t) = Au(t) + f (t, u(t)) t ∈ [0, T ] u(0) = u 0 , D α u(0) = u 1 u (α) (t) = Au(t) + f (t, u(t)) t ∈ [0, T ] u(0) = u 0 , D α u(0) = u 1 Where α ∈ (1, 2], and we use the conformable derivative. We give the notion of α-Cosine families and proveded the existence and uniqueness of the problem 0.1.

Symmetry, 2022

The theory of symmetry has a significant influence in many research areas of mathematics. The cla... more The theory of symmetry has a significant influence in many research areas of mathematics. The class of symmetric functions has wide connections with other classes of functions. Among these, one is the class of convex functions, which has deep relations with the concept of symmetry. In recent years, the Schur convexity, convex geometry, probability theory on convex sets, and Schur geometric and harmonic convexities of various symmetric functions have been extensively studied topics of research in inequalities. The present attempt provides novel portmanteauHermite–Hadamard–Jensen–Mercer-type inequalities for convex functions that unify continuous and discrete versions into single forms. They come as a result of using Riemann–Liouville fractional operators with the joint implementations of the notions of majorization theory and convex functions. The obtained inequalities are in compact forms, containing both weighted and unweighted results, where by fixing the parameters, new and old v...