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Fractional calculus is extremely important and should not be undervalued due to its critical role in the theory of inequalities. In this article, different generalized Hermite-Hadamard type inequalities for functions whose modulus of first derivatives are [Formula: see text]-convex are presented, via Caputo-Fabrizio integrals. Graphical justifications of main results are presented. Graphs enable us to support our conclusions and show the reliability of our findings. Additionally, some applications to probability theory and numerical integration are also established. As special cases, certain established outcomes from different articles are recaptured. This study acts as a stimulant for future studies, inspiring researchers to investigate more thorough results by utilizing generalized fractional operators and broadening the idea of convexity.

The precise solutions of the (2+1)-dimensional modified KdV-KP equation are examined in this study. Three reliable methods, namely, the modified auxiliary equation scheme, extended rational sine–cosine methodology, and extended rational sinh–cosh technique, are utilized for the first time to extract multiple soliton solutions for the governing model. These types of solutions have applications in plasma, solid state, neuronal, biological production, and diffusion processes. Graphical simulations of the obtained solutions are included in the form of 3 dimensional surface plots to reveal their physical structure and dynamical features.

The B-spline or the basis spline function, is a piecewise polynomial function made up of polynomials, its domain is subdivided by knots, and basis functions are non-zero on the entire domain. In this study, a new cubic B-spline (NCBS) approximation together with the θ-weighted scheme is formed to approximate the numerical solution of the time fractional advection diffusion equation (TFADE) involving the Atangana–Baleanu time fractional derivative (ABTFD). The finite difference scheme (FDS) is utilized to discretize the ABTFD with a non-singular kernel. The spatial derivative is discretized by using NCBS functions. The stability analysis of the proposed technique is studied. Convergence analysis of the current technique is also analyzed. The proposed technique is examined on a variety of problems, and the numerical outcomes are contrasted with the previously published technique’s results to ensure the correctness and accuracy of the current technique.