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Differential equations (DEs) performed a vital role in the implementation of almost all the mechanical, physical, or biological processes. Higher order DEs had always been challenging to solve for the researchers so numerous numerical techniques were developed to attain the vital numerical approximations of such types of problems. In this work, highly advanced numerical techniques are established for the approximation of the fourteenth (14th)-order boundary value problems using Adomian decomposition method. The mathematical outcomes of the equations are attained in the form of convergent series that have effortlessly assessable components having step size h = 10. Some numerical examples are also deliberated to demonstrate the capability and application of the established procedure.

The extremely accurate findings of ninth-order linear and nonlinear BVPs are described in this paper. CNPS and CPS are used to find out the numerical roots of linear and nonlinear, ninth-order BVPs. The proposed methods transform the ninth-order BVPs into a system of linear equations. The algorithms developed in this study not only provide approximate solutions to the BVPs using CNPS and CPS, but they also estimate the derivatives from the first to the ninth-order of the analytical solution simultaneously. These approaches are implemented across five diverse problems, showcasing the effectiveness of these techniques by utilizing step sizes of h = 1 / 1 0 and h = 1 / 5. In order to gauge the accuracy of the methods, a comparison is drawn between the outcomes and the exact solutions, which are then presented in tabular and graphical format. The precision of these techniques is demonstrated through a detailed investigation and is found to be superior to the CBS, the Petrov–Galerkin Method (PGM) using Splines functions as basis functions and Septic B Splines functions as weight functions, the collocation method for ninth-order BVPs by Quintic B Splines and Sextic B Splines as evidenced by the comparison of AEs of CPS and CNPS with these alternative methods. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to explore novel variants for convex functions via discrete ℬ -fractional operator in the frame of time scale calculus ℏ ℤ with 0 < α < 1 and 0 < ℏ ≤ 1. Based on a comparison with the integral inequalities, we have provided new discrete fractional inequalities having ℏ -discrete generalized Mittag-Leffler function in the kernel which generates several known results and can be utilized as handy tools in the investigation of qualitative and quantitative properties of solutions of certain classes of difference equations. Several new special cases are also apprehended in the setting of time scale ℤ and . As the application aspect, we provide an illustrated example to show the effectiveness of our new criteria in fractional ℏ -difference type initial value problem. [ABSTRACT FROM AUTHOR]

In this paper, cubic polynomial and nonpolynomial splines are developed to solve solutions of 10th- and 12th-order nonlinear boundary value problems (BVPs). Such types of BVPs occur when a consistent magnetized force field is applied crosswise the fluid in the substance of gravitational force. We will amend our problem into such a form that converts the system of 1 0 th- & 1 2 th-order BVPs into a new system of 2 nd-order BVPs. The appropriate outcomes by using CP Spline and CNP Spline are compared with the exact root. To show the efficiency of our results, absolute errors calculated by using CP Spline and CNP Spline have been compared with other methods like differential transform method, Adomian decomposition method, variational iteration method, cubic B-spline, homotopy perturbation method, 5 th- and 6 th-order B-spline and our results are very encouraging. Graphs and tables are also presented in the numerical section of this paper. [ABSTRACT FROM AUTHOR]

In this paper, some attempts have been devoted to investigating the dynamic features of discrete fractional calculus (DFC). To date, discrete fractional systems with complex dynamics have attracted the most consideration. By considering discrete ℏ -proportional fractional operator with nonlocal kernel, this study contributes to the major consequences of the certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete ℏ -proportional fractional sums, as presented. The proposed system has an intriguing feature not investigated in the literature so far, it is characterized by the nabla ℏ fractional sums. Novel special cases are reported with the intention of assessing the dynamics of the system, as well as to highlighting the several existing outcomes. In terms of applications, we can employ the derived consequences to investigate the existence and uniqueness of fractional difference equations underlying worth problems. Finally, the projected method is efficient in analyzing the complexity of the system. [ABSTRACT FROM AUTHOR]

Lovastatin has received interest for its potential therapeutic use in treating numerous diseases, for example, the blood cholesterol level by restraining hydroxy-3-methylglutaryl coenzyme A (HMG-CoA) reductase. The research utilized the fungal growth bioassay technique to disengage and evaluate filamentous organism for the lovastatin creation. The clever type of Aspergillus terreus (KF971363.1) was embraced for lovastatin creation by solid-state fermentation (SSF). Lovastatin production was optimized using physiological parameters such as pH and temperature at SSF. The addition of nitrogen source enhanced the production of lovastatin by the breakdown of lignocellulose that improved the production of lovastatin. The research verified a yeast growth inhibition bioassay approach, in addition to thin-layer chromatography and liquid chromatography-mass spectrometry (LC-MS). All of these techniques were used to confirm lovastatin production. The purified extract subjected to the TLC analysis showed retention factor (Rf) value of 0.73. Moreover, the inhibition bioassay method reassures the lovastatin production by comparing the zone of inhibition against C. albicans.