In this article, we study an inverse fractional problem that is ill-posed in the sense of Hadamard. The goal is to recover a source term in a time-fractional parabolic equation with a time-dependent coefficient, using Tikhonov regularization. A no-regret control approach is employed to address the regularization problem when the initial condition is unknown. The source term is characterized through the corresponding optimality system.
There are various known refinements to Becker-Stark inequality. In this paper, we present a more rigorous refinement of Becker-Stark inequality which is based on the result of Zhu [15]. Zhu gave a double inequality with quadratic estimations appear on both the left and right sides. In this paper, based on the inequality given by Zhu, we establish a new double inequality with quadratic estimations. The double inequality can be said to be an improvement of Zhu's inequality, as the left side is a larger function and the right side is a smaller function than Zhu's ones.
Redheffer's inequality and its generalization are applied to the study of behavior and estimates of the first eigenvalue of p-Laplacian with respect to p. Furthermore, a Redheffer-type inequality for the generalized trigonometric function is extended to a broader class.
The aim of this paper is to apply the Norlund sum to Bessel function and generalized Bernoulli polynomials involving this function, we derive some new relations and integral formulas. By applying the Laplace transform with its inverse to generating function for the generalized Bernoulli polynomials involving the Bessel function, we also derive infinite series representation for these polynomials. Furthermore, some relationship between the Bessel function and the Norlund sum is found. We also give some new relations by comparing specific values to the Bessel function and modified Bessel functions with the aid of their Laplace transforms by its inverse.
The aim of this paper is to establish new Stirling-type approximation formulas in terms of the digamma and trigamma functions. Our new formulas offer approximations of order n(-2), more accurate than the classical Stirling approximation of order n(-1). Finally, some inequalities are given.
