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Many attempts have been made in recent decades to integrate machine learning (ML) and topological data analysis. A prominent problem in applying persistent homology to ML tasks is finding a vector representation of a persistence diagram (PD), which is a summary diagram for representing topological features. From the perspective of data fitting, a stable vector representation, namely, persistence B-spline grid (PBSG), is developed based on the efficient technique of progressive-iterative approximation for least-squares B-spline function fitting. We theoretically prove that the PBSG method is stable with respect to the metric of 1-Wasserstein distance defined on the PD space. The developed method was tested on a synthetic data set, data sets of randomly generated PDs, data of a dynamical system, and 3D CAD models, showing its effectiveness and efficiency.

This educational paper aims to introduce RDBLS2D, a 99-line MATLAB code that integrates the advantages of the B-spline-based level set function and the reaction–diffusion update scheme to achieve efficient and effective structural topology optimization. It uses 66 and 33 lines for the main program and the B-spline level set representation method updated by the reaction–diffusion scheme, respectively. Also, repeated B-spline’s knots on the boundary of the design domain were used in the code, which can naturally guarantee the connection between neighboring cells in the design of functionally graded materials. Numerical examples of mean compliance minimization demonstrate that the algorithm is capable of producing refined structural profiles featuring exceptionally smooth boundaries in just 20 iteration steps.

The present paper deals with cubic B-spline approximation together with θ-weighted scheme to obtain numerical solution of the time fractional advection diffusion equation using Atangana–Baleanu derivative. To discretize the Atangana–Baleanu time derivative containing a non-singular kernel, finite difference scheme is utilized. The cubic basis functions are associated with spatial discretization. The current discretization scheme used in the present study is unconditionally stable and the convergence is of order O(h2+Δt2). The proposed scheme is validated through some numerical examples which reveal the current scheme is feasible and quite accurate.

We present a collocation approach based on redefined cubic B-spline (RCBS) functions and finite difference formulation to study the approximate solution of time fractional Allen–Cahn equation (ACE). We discretize the time fractional derivative of order α∈(0,1] by using finite forward difference formula and bring RCBS functions into action for spatial discretization. We find that the numerical scheme is of order O(h2+Δt2−α) and unconditionally stable. We test the computational efficiency of the proposed method through some numerical examples subject to homogeneous/nonhomogeneous boundary constraints. The simulation results show a superior agreement with the exact solution as compared to those found in the literature.

In this paper, the vector extension operation is proposed to replace the de Boor‐Cox formula for a fast algorithm to B‐spline basis functions. This B‐spline basis function based on vector extending operation is implemented in the class and shape transformation (CST) parameterization method in place of the traditional Bézier polynomials to enhance the local ability of control and accuracy to represent an airfoil shape. To calculate the k‐degree B‐spline function's nonzero values, the algorithm can improve the computing efficiency by 2k+1 times. Using B‐spline to improve the CST parameterization algorithm increases the parameterization design space and improves the local control ability of the CST algorithm. The vector expansion method is used to realize the fast evaluation of B‐spline, and the calculation efficiency is increased by 2k+1$$ 2k+1 $$ times

In this research, a groundbreaking framework for the octic B-spline collocation method in n-dimensional spaces is presented. This work is an extension of previous works that involved the creation of B-spline functions in n-dimensional space for the purpose of solving mathematical models in n-dimensions. The octic B-spline collocation approach in n-dimensional space is an extension of the standard B-spline collocation approach to higher dimensions. It involves using eighth order (octic) B-splines, which have higher smoothness and continuity properties than lower-order B-splines. To demonstrate the effectiveness and precision of the suggested method, a selection of test problems in two- and three-dimensional space is utilized. For making comparisons, we make use of a wide variety of numerical problems, which are described in this paper.

This paper proposes two numerical approaches for solving the coupled nonlinear time-fractional Burgers’ equations with initial or boundary conditions on the interval [0,L]. The first method is the non-polynomial B-spline method based on L1-approximation and the finite difference approximations for spatial derivatives. The method has been shown to be unconditionally stable by using the Von-Neumann technique. The second method is the shifted Jacobi spectral collocation method based on an operational matrix of fractional derivatives. The proposed algorithms’ main feature is that when solving the original problem it is converted into a nonlinear system of algebraic equations. The efficiency of these methods is demonstrated by applying several examples in time-fractional coupled Burgers equations. The error norms and figures show the effectiveness and reasonable accuracy of the proposed methods.

Purpose The purpose of this paper is to present a new cubic B-spline (CBS) approximation technique for the numerical treatment of coupled viscous Burgers’ equations arising in the study of fluid dynamics, continuous stochastic processes, acoustic transmissions and aerofoil flow theory. Design/methodology/approach The system of partial differential equations is discretized in time direction using the finite difference formulation, and the new CBS approximations have been used to interpolate the solution curves in the spatial direction. The theoretical estimation of stability and uniform convergence of the proposed numerical algorithm has been derived rigorously. Findings A different scheme based on the new approximation in CBS functions is proposed which is quite different from the existing methods developed (Mittal and Jiwari, 2012; Mittal and Arora, 2011; Mittal and Tripathi, 2014; Raslan et al., 2017; Shallal et al., 2019). Some numerical examples are presented to validate the performance and accuracy of the proposed technique. The simulation results have guaranteed the superior performance of the presented algorithm over the existing numerical techniques on approximate solutions of coupled viscous Burgers’ equations. Originality/value The current approach based on new CBS approximations is novel for the numerical study of coupled Burgers’ equations, and as far as we are aware, it has never been used for this purpose before.

In this study, we have proposed an efficient numerical algorithm based on third degree modified extended B-spline (EBS) functions for solving time-fractional diffusion wave equation with reaction and damping terms. The Caputo time-fractional derivative has been approximated by means of usual finite difference scheme and the modified EBS functions are used for spatial discretization. The stability analysis and derivation of theoretical convergence validates the authenticity and effectiveness of the proposed algorithm. The numerical experiments show that the computational outcomes are in line with the theoretical expectations. Moreover, the numerical results are proved to be better than other methods on the topic.