Some recent results have been found treating the famous Simpson’s rule in connection with the convexity property of functions and those called generalized convex. The purpose of this article is to address Newton-type integral inequalities by associating with them certain criteria of quantum calculus and the convexity of the functions of various variables. In this article, by using the concept of recently defined $q_1{q}_2$ -derivatives and integrals, some of Newton’s type inequalities for co-ordinated convex functions are revealed. We also employ the limits of $q_1,q_2\to 1^{-}$ in new results, and attain some new inequalities of Newton’s type for co-ordinated convex functions through ordinary integral. Finally, we provide a thorough application of the newly obtained key outcomes, these new consequences can be useful in the integral approximation study for symmetrical functions, or with some kind of symmetry.

In this investigation, for convex functions, some new $(p,q)$–Hermite–Hadamard-type inequalities using the notions of ${(p,q)}^{{\pi }_2}$ derivative and ${(p,q)}^{{\pi }_2}$ integral are obtained. Furthermore, for ${(p,q)}^{{\pi }_2}$-differentiable convex functions, some new ($p,q$) estimates for midpoint and trapezoidal-type inequalities using the notions of ${(p,q)}^{{\pi }_2}$ integral are offered. It is also shown that the newly proved results for $p=1$ and $q\to 1^{-}$ can be converted into some existing results. Finally, we discuss how the special means can be used to address newly discovered inequalities.

Integral inequality is an interesting mathematical model due to its wide and significant applications in mathematical analysis and fractional calculus. In the present research article, we obtain new inequalities of Simpson’s integral type based on theφ-convex andφ-quasiconvex functions in the second derivative sense. In the last sections, some applications on special functions are provided and shown via two figures to demonstrate the explanation of the readers.

A non-polynomial spline is a technique that utilizes information from symmetric functions to solve mathematical or physical models numerically. This paper introduces a novel non-polynomial spline construct incorporating a rational function term to develop an efficient numerical scheme for solving time-fractional differential equations. The proposed method is specifically applied to the time-fractional KdV–Burgers (TFKdV) equation. and time-fractional differential equations are crucial in physics as they provide a more accurate description of various complex processes, such as anomalous diffusion and wave propagation, by capturing memory effects and non-local interactions. Using Taylor expansion and truncation error analysis, the convergence order of the numerical scheme is derived. Stability is analyzed through the Fourier stability criterion, confirming its conditional stability. The accuracy and efficiency of the rational non-polynomial spline (RNPS) method are validated by comparing numerical results from a test example with analytical and previous solutions, using norm errors. Results are presented in $2D$ and $3D$ graphical formats, accompanied by tables highlighting performance metrics. Furthermore, the influences of time and the fractional derivative are examined through graphical analysis. Overall, the RNPS method has demonstrated to be a reliable and effective approach for solving time-fractional differential equations.