This study explores the fractional damped generalized regularized long‐wave equation in the sense of Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio fractional derivatives. With the aid of fixed‐point theorem in the Atangana‐Baleanu fractional derivative with Mittag‐Leffler–type kernel, we show the existence and uniqueness of the solution to the damped generalized regularized long‐wave equation. The modified Laplace decomposition method (MLDM) defined in the sense of Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio (in the Riemann sense) operators is used in securing the approximate‐analytical solutions of the nonlinear model. The numerical simulations of the obtained solutions are performed with different suitable values of ρ, which is the order of fractional parameter. We have seen the effect of the various parameters and variables on the displacement in figures.

The main issues addressed in this paper are making generalization of Gronwall, Volterra and Pachpatte type inequalities for conformable differential equations. By using the Katugampola definition for conformable calculus we found some upper and lower bound for integral inequalities. The established results are extensions of some existing Gronwall, Volterra and Pachpatte type inequalities in the previous published studies.

In this study we introduced and tested retarded conformable fractional integral inequalities utilizing non-integer order derivatives and integrals. In line with this purpose, we used the Katugampola type conformable fractional calculus which has several practical properties. These inequalities generalize some famous integral inequalities which provide explicit bounds on unknown functions. The results provided here had been implemented to the global existence of solutions to the conformable fractional differential equations with time delay.

By using contemporary theory of inequalities, this study is devoted to propose a number of refinements inequalities for the Hermite−Hadamard's type inequality and conclude explicit bounds for the trapezoid inequalities in terms of s-convex mappings, at most second derivative through the instrument of generalized fractional integral operator and a considerable amount of results for special means. The results of this study which are the generalization of those given in earlier works are obtained for functions f where | f | and | f | (or | f | q and | f | q for q ≥ 1) are s-convex hold by applying the Hölder inequality and the power mean inequality.

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