Abstract:In recent times, numerical approximation of 3rd-order boundary value problems (BVPs) has attracted great attention due to its wide applications in solving problems arising from sciences and engineering. Hence, A higher-order block method is constructed for the direct solution of 3rd-order linear and non-linear BVPs. The approach of interpolation and collocation is adopted in the derivation. Power series approximate solution is interpolated at the points required to suitably handle both linear and non-linear th… Show more
“…The permutation problem is a serious obstacle for calculating thirdorder linking numbers. A requirement for higher-order winding numbers is to avoid such problems [11]. In this paper, we are showing that the Evans-Berger and Akhmetev's formulas coincide in certain cases.…”
In this work, we demonstrate that the integral formula for a generalised Sato-Levine invariant is consistent in certain situations with Evans and Berger's formula for the fourth-order winding number. Also, we found that, in principle, one can derive analogous high-order winding numbers by which one can calculate the entanglement of braids. The winding number for the Brunnian 4-braid is calculated algebraically using the cup product on the cohomology of a finite regular CW-space which is the complement $\mathbb{R}^3\backslash \mathcal{B}_4$.
“…The permutation problem is a serious obstacle for calculating thirdorder linking numbers. A requirement for higher-order winding numbers is to avoid such problems [11]. In this paper, we are showing that the Evans-Berger and Akhmetev's formulas coincide in certain cases.…”
In this work, we demonstrate that the integral formula for a generalised Sato-Levine invariant is consistent in certain situations with Evans and Berger's formula for the fourth-order winding number. Also, we found that, in principle, one can derive analogous high-order winding numbers by which one can calculate the entanglement of braids. The winding number for the Brunnian 4-braid is calculated algebraically using the cup product on the cohomology of a finite regular CW-space which is the complement $\mathbb{R}^3\backslash \mathcal{B}_4$.
“…According to the established theorem by in Familua et al and Jain et al in references [43] and [57]), this subsection analyzed the order of accuracy, constants of errors, zero-stability, and finally the consistency of the ISBS. The linear operator listed below can be used to represent the Scheme (28) and its related variants.…”
Section: Preliminary Of Isbs's Theoretical Analysismentioning
confidence: 99%
“…where � C s ; s ¼ 1; 2; ::: Definition 3.1.1 (Order) (Familua et al [43]) The ISBS (29) and its linear operators are assigned an order p if The inverse of a matrix A denoted by A −1 is given as A À 1 ¼ adjðAÞ jAj . where adjðAÞ is the ad joint of matrix A and |A| is the determinant of the matrix.…”
Section: Preliminary Of Isbs's Theoretical Analysismentioning
confidence: 99%
“…Olaiya et al [ 42 ] scrutinized the numerical models utilized in resolving the solutions for the Black-Scholes differential equation. Additionally, Familua et al [ 43 ] conducted an in-depth examination of advanced self-starting algorithms tailored for numerically simulating differential equations featuring second derivatives, offering diverse practical applications.…”
In the era of computational advancements, harnessing computer algorithms for approximating solutions to differential equations has become indispensable for its unparalleled productivity. The numerical approximation of partial differential equation (PDE) models holds crucial significance in modelling physical systems, driving the necessity for robust methodologies. In this article, we introduce the Implicit Six-Point Block Scheme (ISBS), employing a collocation approach for second-order numerical approximations of ordinary differential equations (ODEs) derived from one or two-dimensional physical systems. The methodology involves transforming the governing PDEs into a fully-fledged system of algebraic ordinary differential equations by employing ISBS to replace spatial derivatives while utilizing a central difference scheme for temporal or y-derivatives. In this report, the convergence properties of ISBS, aligning with the principles of multi-step methods, are rigorously analyzed. The numerical results obtained through ISBS demonstrate excellent agreement with theoretical solutions. Additionally, we compute absolute errors across various problem instances, showcasing the robustness and efficacy of ISBS in practical applications. Furthermore, we present a comprehensive comparative analysis with existing methodologies from recent literature, highlighting the superior performance of ISBS. Our findings are substantiated through illustrative tables and figures, underscoring the transformative potential of ISBS in advancing the numerical approximation of two-dimensional PDEs in physical systems.
“…Likewise, Olaiya et al in [34] presented a numerical approach for simulating the Black-Scholes partial differential equation via a two-step off-grip block of algorithms of algebraic order seven. Lastly, the work of Familua et al in [35] considered a higher-order block technique for the numerical simulation of thirdorder boundary value problems with applications. The theoretical analysis of the methods was investigated and discussed comprehensively.…”
A computational approach with the aid of the Linear Multistep Method (LMM) for the numerical solution of differential equations with initial value problems or boundary conditions has appeared several times in the literature due to its good accuracy and stability properties. The major objective of this article is to extend a multistep approach for the numerical solution of the Partial Differential Equation (PDE) originating from fluid mechanics in a two-dimensional space with initial and boundary conditions, as a result of the importance and utility of the models of partial differential equations in applications, particularly in physical phenomena, such as in convection-diffusion models, and fluid flow problems. Thus, a multistep collocation formula, which is based on orthogonal polynomials is proposed. Ninth-order Multistep Collocation Formulas (NMCFs) were formulated through the principle of interpolation and collocation processes. The theoretical analysis of the NMCFs reveals that they have algebraic order nine, are zero-stable, consistent, and, thus, convergent. The implementation strategies of the NMCFs are comprehensively discussed. Some numerical test problems were presented to evaluate the efficacy and applicability of the proposed formulas. Comparisons with other methods were also presented to demonstrate the new formulas’ productivity. Finally, figures were presented to illustrate the behavior of the numerical examples.
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