In this paper, we establish some new Milne?s type inequalities for the differentiable convex functions in quantum calculus (q-calculus). We prove q-integral identity first and then we prove some new Milne?s type inequalities for q-differentiable convex functions. These inequalities play an important role in Open-Newton?s Cotes formulas. Furthermore, we give the computational analysis of these inequalities for convex functions and prove that the bounds of this paper are better than the existing ones. Ultimately, we provide some mathematical examples to show the validity of newly establish inequalities in q-calculus.

Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus.

In this work, we prove a new $(p,q)$-integral identity involving a $(p,q)$-derivative and $(p,q)$-integral. The newly established identity is then used to show some new Simpson’s formula type inequalities for $(p,q)$-differentiable convex functions. Finally, the newly discovered results are shown to be refinements of comparable results in the literature. Analytic inequalities of this type, as well as the techniques used to solve them, have applications in a variety of fields where symmetry is important.

This work investigates solitary wave solutions and dynamical properties of the integrable Zhanbota‐IIA equation, which exhibits rich nonlinear dynamics and diverse soliton structures. To derive exact traveling wave solutions, two robust analytical frameworks are employed: the new extended direct algebraic method (NEDAM) and the ( G ^′ / G ² )‐expansion method. These methods generate solutions in the forms of trigonometric, hyperbolic, and rational functions, yielding diverse wave profiles, including bright and dark solitons, periodic and mixed periodic forms, parabolic waves, and shock wave profiles. A Galilean transformation is applied to reduce the equation to its associated dynamical system, enabling a comprehensive exploration of wave propagation dynamics. Sensitivity and chaotic behaviors are analyzed through phase portraits, time series, return maps, Lyapunov exponents, and parameter perturbations under elliptic forcing, revealing pronounced sensitivity to initial conditions, multistability, and complex nonlinear behavior. These findings provide valuable insights into the stability properties of the underlying physical systems and demonstrate the robustness of the proposed analytical frameworks for solving complex nonlinear partial differential equations.