Use of Cubic B-Spline in Approximating Solutions of Boundary Value Problems

Here we investigate the use of cubic B-spline functions in solving boundary value problems. First, we derive the linear, quadratic, and cubic B-spline functions. Then we use the cubic B-spline functions to solve second order linear boundary value problems. We consider constant coefficient and variable coefficient cases with non-homogeneous boundary conditions for ordinary differential equations. We also use this numerical method for the space variable to obtain solutions for second order linear partial differential equations. Numerical results for various cases are presented and compared with exact solutions

Use of Cubic B-Spline in Approximating Solutions of Boundary Value Problems

Here we investigate the use of cubic B-spline functions in solving boundary value problems. First, we derive the linear, quadratic, and cubic B-spline functions. Then we use the cubic B-spline functions to solve second order linear boundary value problems. We consider constant coefficient and variable coefficient cases with non-homogeneous boundary conditions for ordinary differential equations. We also use this numerical method for the space variable to obtain solutions for second order linear partial differential equations. Numerical results for various cases are presented and compared with exact solutions

A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability: generalized Korteweg-de Vries equation

This work deals with the constitute of numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with Petrov-Galerkin finite element approach utilising a cubic B-spline function as the trial function and a quadratic function as the test function. Accurateness and effectiveness of the submitted methods are shown by employing propagation of single solitary wave. The L2, L∞error norms and I1, I2and I3invariants are used to validate the applicability and durability of our numerical algorithm. Implementing the Von-Neumann theory, it is manifested that the suggested method is marginally stable. Furthermore, supernonlinear traveling wave solution of the GKdV equation is presented using phase plots. It is seen that the GKdV equation supports superperiodic traveling wave solution only and it is significantly affected by velocity and nonlinear parameters. Also, considering a superficial periodic forcing multistability of traveling waves of perturbed GKdV equation is presented. It is found that the perturbed GKdV equation supports coexisting chaotic and various quasiperiodic features with same parametric values at different initial condition

Active Optimal Control of the KdV Equation Using the Variational Iteration Method

The optimal pointwise control of the KdV equation is investigated with an objective of minimizing a given performance measure. The performance measure is specified as a quadratic functional of the final state and velocity functions along with the energy due to open- and closed-loop controls. The minimization of the performance measure over the controls is subjected to the KdV equation with periodic boundary conditions and appropriate initial condition. In contrast to standard optimal control or variational methods, a direct control parameterization is used in this study which presents a distinct approach toward the solution of optimal control problems. The method is based on finite terms of Fourier series approximation of each time control variable with unknown Fourier coefficients and frequencies. He's variational iteration method for the nonlinear partial differential equations is applied to the problem and thus converting the optimal control of lumped parameter systems into a mathematical programming. A numerical simulation is provided to exemplify the proposed method

Total positivity and accurate computations with Gram matrices of Said-Ball bases

In this article, it is proved that Gram matrices of totally positive bases of the space of polynomials of a given degree on a compact interval are totally positive. Conditions to guarantee computations to high relative accuracy with those matrices are also obtained. Furthermore, a fast and accurate algorithm to compute the bidiagonal factorization of Gram matrices of the Said-Ball bases is obtained and used to compute to high relative accuracy their singular values and inverses, as well as the solution of some linear systems associated with these matrices. Numerical examples are included

Accurate and efficient computations with Wronskian matrices of Bernstein and related bases

In this article, we provide a bidiagonal decomposition of the Wronskian matrices of Bernstein bases of polynomials and other related bases such as the Bernstein basis of negative degree or the negative binomial basis. The mentioned bidiagonal decompositions are used to achieve algebraic computations with high relative accuracy for these Wronskian matrices. The numerical experiments illustrate the accuracy obtained using the proposed decomposition when computing inverse matrices, eigenvalues or singular values, and the solution of some related linear systems. © 2021 John Wiley & Sons Ltd

李群在数值求解偏微分方程中的新应用

随着计算机技术的提高及数值计算方法的不断完善，数值计算逐渐成为研究海洋工程领域内水波动力学的有效手段，以前难以处理的非线性现象研究课题在数值手段的帮助下也出现了求解的可能。本文所研究的李群理论在非线性偏微分方程数值求解中的应用，便是这些手段中的一种。本文的选题背景为纯粹数学理论的应用与研究，通过对非线性力学方程求解方法研究现状的分析，结合李群在偏微分方程中的应用理论，提出一种偏微分方程的降维方法：从无穷小变换出发，构建保持偏微分方程不变性的李群，导出偏微分方程的降维系统，结合非经典方法中的解曲面条件，避开群不变量解的直接求解，同时实现偏微分方程的降维简化，得出原偏微分方程的数值描述。在此基础上，本文在应用实例中通过与某些偏微分方程已知精确解的比较，结果验证了该方法的合理性及有效性。在利用该方法求解几个典型的非线性水波动力学偏微分方程时，得出了一系列关于这些方程的新的数值解，并由此揭示了一些有意义的水波物理现象。 本文研究工作的意义在于丰富了偏微分方程的数值解法，为利用数值方法处理非线性问题提供了新思路，通过对水波动力学方程的求解，为海洋潮流、波浪能的基础研究做出了必要的理论铺垫，对海洋工程的实际应用也具有一定的参考价值