Effective Role of Cubic Splines for the Numerical Computations of Non-Linear Model of Viscoelastic Fluid - Khalid, Aasma; Batool, Rubab; Abro, Kashif Ali; Rehan, Akmal (Acta Mechanica et Automatica, 2025)
The seventh-order boundary value problems (BVPs), which are important because of their complexity and prevalence in many scientific and engineering fields, are the subject of this paper’s study. These high-order boundary value problems appear in fields such as fluid dynamics, where they are used to model fluid flow, and in elasticity theory, where they help describe the deformation of materials. Unfortunately, the precision and stability required to solve these high-order problems consistently are frequently lacking from current numerical techniques. Consequently, the advancement of theoretical research as well as practical applications in these disciplines depends on the development of a reliable and accurate method for solving seventh-order boundary value problems. In order to improve the accuracy and stability of solutions for these challenging issues, we propose novel numerical strategies that involves non-polynomial and polynomial cubic splines. For both methods, the domain [0,1] is divided into sub-intervals with step sizes of h=1/10 and h=1/5. This method involves initially transforming the seventh-order boundary value problems into a system of second-order. These second-order boundary value problems are then discretized using finite difference approximations, incorporating essential boundary conditions, and ultimately converted into a set of linear algebraic equations. The employed methods are rigorously assessed through experimentation on three distinct test problems. The outcomes attained showcase an exceptional level of accuracy, extending up to 7 decimal places. These commendable results are vividly depicted in both the tabulated data and accompanying graphs. Such a high degree of precision substantiates the dependability and efficiency of the proposed method. Comparisons, presented in tables and graphs, highlight the precision and reliability of our methods. These comparisons confirm that our approaches are valuable tools for addressing the challenges associated with seventh-order boundary value problems, marking a notable contribution to the field of numerical analysis. While lower-order boundary value problems have been extensively studied, applying these splines methods to seventh-order boundary value problems presents new challenges and insights. The novelty of this work involves non-polynomial and polynomial cubic spline techniques to solve seventh-order boundary value problems, offering improved accuracy and stability over existing numerical methods.