This paper is devoted to the study of the Cubic B-splines to find the numerical solution of linear and non-linear 8th order BVPs that arises in the study of astrophysics, magnetic fields, astronomy, beam theory, cylindrical shells, hydrodynamics and hydro-magnetic stability, engineering, applied physics, fluid dynamics, and applied mathematics. The recommended method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 8th order BVPs using Cubic-B spline but it also describes the estimated derivatives of 1st order to 8th order of the analytic solution. The strategy is effectively applied to numerical examples and the outcomes are compared with the existing results. The method proposed in this paper provides better approximations to the exact solution.

The present paper is related to the analytical solutions of some heat like equations, using a novel approach with Caputo operator. The work is carried out mainly with the use of an effective and straight procedure of the Iterative Laplace transform method. The proposed method provides the series form solution that has the desired rate of convergence towards the exact solution of the problems. It is observed that the suggested method provides closed-form solutions. The reliability of the method is confirmed with the help of some illustrative examples. The graphical representation has been made for both fractional and integer-order solutions. Numerical solutions that are in close contact with the exact solutions to the problems are investigated. Moreover, the sample implementation of the present method supports the importance of the method to solve other fractional-order problems in sciences and engineering.

By using the generalized beta function, we extend the fractional derivative operator of the Riemann-Liouville and discusses its properties. Moreover, we establish some relations to extended special functions of two and three variables via generating functions.

A new class of 2m-point non-stationary subdivision schemes (SSs) is presented, including some of their important properties, such as continuity, curvature, torsion monotonicity, and convexity preservation. The multivariate analysis of subdivision schemes is extended to a class of non-stationary schemes which are asymptotically equivalent to converging stationary or non-stationary schemes. A comparison between the proposed schemes, their stationary counterparts and some existing non-stationary schemes has been depicted through examples. It is observed that the proposed SSs give better approximation and more effective results.

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<abstract><p>In this research, the Shehu transform is coupled with the Adomian decomposition method for obtaining the exact-approximate solution of the plasma fluid physical model, known as the Zakharov-Kuznetsov equation (briefly, ZKE) having a fractional order in the Caputo sense. The Laplace and Sumudu transforms have been refined into the Shehu transform. The action of weakly nonlinear ion acoustic waves in a plasma carrying cold ions and hot isothermal electrons is investigated in this study. Important fractional derivative notions are discussed in the context of Caputo. The Shehu decomposition method (SDM), a robust research methodology, is effectively implemented to generate the solution for the ZKEs. A series of Adomian components converge to the exact solution of the assigned task, demonstrating the solution of the suggested technique. Furthermore, the outcomes of this technique have generated important associations with the precise solutions to the problems being researched. Illustrative examples highlight the validity of the current process. The usefulness of the technique is reinforced via graphical and tabular illustrations as well as statistics theory.</p></abstract>

A user-friendly approach depending on nonlocal kernel has been constituted in this study to model nonlocal behaviors of fractional differential and difference equations, which is known as a generalized proportional fractional operator in the Hilfer sense. It is deemed, for differentiable functions, by a fractional integral operator applied to the derivative of a function having an exponential function in the kernel. This operator generalizes a novel version of Čebyšev-type inequality in two and three variables sense and furthers the result of existing literature as a particular case of the Čebyšev inequality is discussed. Some novel special cases are also apprehended and compared with existing results. The outcome obtained by this study is very broad in nature and fits in terms of yielding an enormous number of relating results simply by practicing the proportionality indices included therein. Furthermore, the outcome of our study demonstrates that the proposed plans are of significant importance and computationally appealing to deal with comparable sorts of differential equations. Taken together, the results can serve as efficient and robust means for the purpose of investigating specific classes of integrodifferential equations.

Subdivision schemes are extensively used in scientific and practical applications to produce continuous shapes in an iterative way. This paper introduces a framework to compute subdivision depths of ternary schemes. We first use subdivision algorithm in terms of convolution to compute the error bounds between two successive polygons produced by refinement procedure of subdivision schemes. Then, a formula for computing bound between the polygon at k-th stage and the limiting polygon is derived. After that, we predict numerically the number of subdivision steps (depths) required for smooth limiting shape based on the demand of user specified error (distance) tolerance. In addition, extensive numerical experiments were carried out to check the numerical outcomes of this new framework. The proposed methods are more efficient than the method proposed by Song et al.