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A stable and accurate finite-difference discretization of first-order elastic wave equations is derived in this work. To simplify the origin and proof of the formulas, a symmetric matrix form (SMF) for elastic wave equations is presented. The curve domain is discretized using summation-by-parts (SBP) operators, and the boundary conditions are weakly enforced using the simultaneous-approximation-term (SAT) technique, which gave rise to a provably stable high-order SBP-SAT method via the energy method. In addition, SMF can be extended to wave equations of different types (SH wave and P-SV wave) and dimensions, which can simplify the boundary derivation process and improve its applicability. Application of this approximation can divide the domain into a multiblock context for calculation, and the interface boundary conditions of blocks can also be used to simulate cracks and other structures. Several numerical simulation examples, including actual elevation within the area of Lushan, China, are presented, which verifies the viability of the framework present in this paper. The applicability of simulating elastic wave propagation and the application potential in the seismic numerical simulation of this method are also revealed.

For many problems in Physics and Computational Fluid Dynamics (CFD), providing an accurate approximation of derivatives is a challenging task. This paper presents a class of high order numerical schemes for approximating the first derivative. These approximations are derived based on solving a special system of equations with some unknown coefficients. The construction method provides numerous types of schemes with different orders of accuracy. The accuracy of each scheme is analyzed by using Fourier analysis, which illustrates the dispersion and dissipation of the scheme. The polynomial technique is used to verify the order of accuracy of the proposed schemes by obtaining the error terms. Dispersion and dissipation errors are calculated and compared to show the features of high order schemes. Furthermore, there is a plan to study the stability and accuracy properties of the present schemes and apply them to standard systems of time dependent partial differential equations in CFD.

A numerical method for the solution of the Falkner–Skan equation, which is a nonlinear differential equation, is presented. The method has been derived by truncating the semi-infinite domain of the problem to a finite domain and then expanding the required approximate solution as the elements of the Chebyshev series. Using matrix representation of a function and their derivatives, the problem is reduced to a system of algebraic equations in a simple way. From the computational point of view, the results are in excellent agreement with those presented in published works.

As one application of the Cauchy–Binet formula, the Ball–Barthe inequality plays a key role in solving reverse isoperimetric inequalities. Recently, a new Grassmannian Ball–Barthe inequality was established. With it, reverse isoperimetric inequalities on Grassmann manifolds were successfully proved. In this paper, we will establish its matrix form with a simpler proof than that of Grassmannian form. The matrix form also recovers the classical Ball–Barthe inequality in a special case. [ABSTRACT FROM AUTHOR]

This paper introduces a 3D surface reconstruction of mathematical modelling by using blossoming, dependent on a parameter in the coefficient of the control points, the bi-cubic B-spline surface in matrix form scheme can be utilized to obtain a better quality of reconstructed surface, by using different values of parameter of coefficients used for controlling generating 3D complex surface, which is not easy to recreate. The de-Boor method is used to upgrade the matrix form of 2D to bicubic 3D- b-spline surface depends on parameter value. The model can be seen more efficient of surface in comparison with that needed in conventional methods. The effectiveness of the proposed algorithm is illustrated through several comparative examples of 3D matrices surface. The change in surface is made without any change in the control points. Applications of the modeling in both the curve and surface can be used in many fields, such as banknote design, shape design, decorations, governmental document, and other documents, all of which are of high impotence. And it is used to find the appropriate solutions that prevent or reduce the forgery and counterfeiting of the important and vital documents