Solutions of differential equations in a Bernstein polynomial basis

AbstractAn algorithm for approximating solutions to differential equations in a modified new Bernstein polynomial basis is introduced. The algorithm expands the desired solution in terms of a set of continuous polynomials over a closed interval and then makes use of the Galerkin method to determine the expansion coefficients to construct a solution. Matrix formulation is used throughout the entire procedure. However, accuracy and efficiency are dependent on the size of the set of Bernstein polynomials and the procedure is much simpler compared to the piecewise B spline method for solving differential equations. A recursive definition of the Bernstein polynomials and their derivatives are also presented. The current procedure is implemented to solve three linear equations and one nonlinear equation, and excellent agreement is found between the exact and approximate solutions. In addition, the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques. This procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations

Application of Zernike polynomials in solving certain first and second order partial differential equations

Integration operational matrix methods based on Zernike polynomials are used to determine approximate solutions of a class of non-homogeneous partial differential equations (PDEs) of first and second order. Due to the nature of the Zernike polynomials being described in the unit disk, this method is particularly effective in solving PDEs over a circular region. Further, the proposed method can solve PDEs with discontinuous Dirichlet and Neumann boundary conditions, and as these discontinuous functions cannot be defined at some of the Chebyshev or Gauss-Lobatto points, the much acclaimed pseudo-spectral methods are not directly applicable to such problems. Solving such PDEs is also a new application of Zernike polynomials as so far the main application of these polynomials seem to have been in the study of optical aberrations of circularly symmetric optical systems. In the present method, the given PDE is converted to a system of linear equations of the form Ax = b which may be solved by both l1 and l2 minimization methods among which the l1 method is found to be more accurate. Finally, in the expansion of a function in terms of Zernike polynomials, the rate of decay of the coefficients is given for certain classes of functions

A fast and well-conditioned spectral method for singular integral equations

We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in ${\cal O}(m^2n)$ operations using an adaptive QR factorization, where $m$ is the bandwidth and $n$ is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to ${\cal O}(m n)$ operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The Julia software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface

Approximate solution of singular integral equations of the first kind with Cauchy kernel

AbstractIn this work a study of efficient approximate methods for solving the Cauchy type singular integral equations (CSIEs) of the first kind, over a finite interval, is presented. In the solution, Chebyshev polynomials of the first kind, Tn(x), second kind, Un(x), third kind, Vn(x), and fourth kind, Wn(x), corresponding to respective weight functions W(1)(x)=(1−x2)−12, W(2)(x)=(1−x2)12,W(3)(x)=(1+x)12(1−x)−12 and W(4)(x)=(1+x)−12(1−x)12, have been used to obtain systems of linear algebraic equations. These systems are solved numerically. It is shown that for a linear force function the method of approximate solution gives an exact solution, and it cannot be generalized to any polynomial of degree n. Numerical results for other force functions are given to illustrate the efficiency and accuracy of the method

Formally biorthogonal polynomials and a look-ahead Levinson algorithm for general Toeplitz systems

Systems of linear equations with Toeplitz coefficient matrices arise in many important applications. The classical Levinson algorithm computes solutions of Toeplitz systems with only O(n(sub 2)) arithmetic operations, as compared to O(n(sub 3)) operations that are needed for solving general linear systems. However, the Levinson algorithm in its original form requires that all leading principal submatrices are nonsingular. An extension of the Levinson algorithm to general Toeplitz systems is presented. The algorithm uses look-ahead to skip over exactly singular, as well as ill-conditioned leading submatrices, and, at the same time, it still fully exploits the Toeplitz structure. In our derivation of this algorithm, we make use of the intimate connection of Toeplitz matrices with formally biorthogonal polynomials

Fast Algorithms for Solving FLS R

Block circulant and circulant matrices have already become an ideal research area for solving various differential equations. In this paper, we give the definition and the basic properties of FLS R-factor block circulant (retrocirculant) matrix over field F. Fast algorithms for solving systems of linear equations involving these matrices are presented by the fast algorithm for computing matrix polynomials. The unique solution is obtained when such matrix over a field F is nonsingular. Fast algorithms for solving the unique solution of the inverse problem of AX=b in the class of the level-2 FLS (R,r)-circulant(retrocirculant) matrix of type (m,n) over field F are given by the right largest common factor of the matrix polynomial. Numerical examples show the effectiveness of the algorithms

New Techniques for Polynomial System Solving

Since any encryption map may be viewed as a polynomial map between finite dimensional vector spaces over finite fields, the security of a cryptosystem can be examined by studying the difficulty of solving large systems of multivariate polynomial equations. Therefore, algebraic attacks lead to the task of solving polynomial systems over finite fields. In this thesis, we study several new algebraic techniques for polynomial system solving over finite fields, especially over the finite field with two elements. Instead of using traditional Gröbner basis techniques we focus on highly developed methods from several other areas like linear algebra, discrete optimization, numerical analysis and number theory. We study some techniques from combinatorial optimization to transform a polynomial system solving problem into a (sparse) linear algebra problem. We highlight two new kinds of hybrid techniques. The first kind combines the concept of transforming combinatorial infeasibility proofs to large systems of linear equations and the concept of mutants (finding special lower degree polynomials). The second kind uses the concept of mutants to optimize the Border Basis Algorithm. We study recent suggestions of transferring a system of polynomial equations over the finite field with two elements into a system of polynomial equalities and inequalities over the set of integers (respectively over the set of reals). In particular, we develop several techniques and strategies for converting the polynomial system of equations over the field with two elements to a polynomial system of equalities and inequalities over the reals (respectively over the set of integers). This enables us to make use of several algorithms in the field of discrete optimization and number theory. Furthermore, this also enables us to investigate the use of numerical analysis techniques such as the homotopy continuation methods and Newton's method. In each case several conversion techniques have been developed, optimized and implemented. Finally, the efficiency of the developed techniques and strategies is examined using standard cryptographic examples such as CTC and HFE. Our experimental results show that most of the techniques developed are highly competitive to state-of-the-art algebraic techniques

Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization

The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.Comment: 28 pages, survey pape