Approximate Polynomial Solutions Research Articles

An efficient spectral method for solving third-kind Volterra integral equations with non-smooth solutions

This paper is concerned with the numerical solution of Volterra integral equations of the third kind with non-smooth solutions based on the recursive approach of the spectral Tau method. To this end, a new set of the fractional version of canonical basis polynomials (called FC-polynomials) is introduced. The approximate polynomial solution (called Tau-solution) is expressed in terms of FC-polynomials. The fractional structure of Tau-solution allows recovering the standard degree of accuracy of spectral methods even in the case of non-smooth solutions. The convergence analysis of the method is carried out. The obtained numerical results show the accuracy and efficiency of the method compared to other existing methods.

A Symbolic-Numeric Validation Algorithm for Linear ODEs with Newton–Picard Method

A symbolic-numeric validation algorithm is developed to compute rigorous and tight uniform error bounds for polynomial approximate solutions to linear ordinary differential equations, and in particular D-finite functions. It relies on an a posteriori validation scheme, where such an error bound is computed afterwards, independently from how the approximation was built. Contrary to Newton–Galerkin validation methods, widely used in the mathematical community of computer-assisted proofs, our algorithm does not rely on finite-dimensional truncations of differential or integral operators, but on an efficient approximation of the resolvent kernel using a Chebyshev spectral method. The result is a much better complexity of the validation process, carefully investigated throughout this article. Indeed, the approximation degree for the resolvent kernel depends linearly on the magnitude of the input equation, while the truncation order used in Newton–Galerkin may be exponential in the same quantity. Numerical experiments based on an implementation in C corroborate this complexity advantage over other a posteriori validation methods, including Newton–Galerkin.

On Convergence of Polynomial Approximate Solutions of Minimal Surface Equations in Domains Satisfying the Cone Condition

In this paper we consider the polynomial approximate solutions of the minimal surface equation. It is shown that under certain conditions on the geometric structure of the domain the absolute values of the gradients of the solutions are bounded as the degree of these polynomials increases. The obtained properties imply the uniform convergence of approximate solutions to the exact solution of the minimal surface equation. In numerical solving of boundary value problems for equations and systems of partial differential equations, a very important issue is the convergence of approximate solutions.The study of this issue is especially important for nonlinear equations since in this case there is a series of difficulties related with the impossibility of employing traditional methods and approaches used for linear equations. At present, a quite topical problem is to determine the conditions ensuring the uniform convergence of approximate solutions obtained by various methods for nonlinear equations and system of equations of variational kind (see, for instance, [1]). For nonlinear equations it is first necessary to establish some a priori estimates of the derivatives of approximate solutions.In this paper, we gave a substantiation of the variational method of solving the minimal surface equation in the case of multidimensional space.We use the same approach that we used in [3] for a two-dimensional equation. Note that such a convergence was established in [3] under the condition that a certain geometric characteristic Δ(Ω) in the domain Ω, in which the solutions are considered, is positive. In particular, domain with a smooth boundary satisfied this requirement. However, this characteristic is equal to zero for a fairly wide class of domains with piecewise-smooth boundaries and sufficiently "narrow" sections at the boundary. For example, such a section of the boundary is the vertex of a cone with an angle less than π/2. In this paper, we present another approach to determining the value of Δ(Ω) in terms of which it is possible to extend the results of the work [3] in domain satisfying cone condition.

A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.

An approximation technique for solutions of singularly perturbed one‐dimensional convection‐diffusion problem

AbstractIn this study, a weighted residual method is presented in order to numerically solve singularly perturbed one‐dimensional parabolic convection‐diffusion problem. Assuming an approximate polynomial solution of a prescribed degree N, the method uses the set of bivariate monomials whose degrees do not exceed N as the set of base functions. Then, following Galerkin's path, inner product with the base functions are applied to the residual of the approximate solution polynomial. Incorporation of the initial and boundary conditions are ensured by forcing the approximate solution to satisfy these conditions on equidistant collocation points. The solution of the resulting linear system then yields the approximate polynomial solution. Additionally, the technique of residual correction, which aims to increase the accuracy of the approximate solution by estimating its error, is discussed briefly. The Galerkin‐like scheme and the residual correction technique are illustrated with two examples. The obtained results are also compared with other methods presented in the literature.

Polynomial solution of singular differential equations using Weighted Sobolev gradients

ABSTRACTIn this work, we proposed a new scheme based on Sobolev gradient approach for finding an approximate polynomial solution of certain first and second order ordinary differential equations. Continuous function instead of discretized differential operator is used to avoid numerical issues posed by the size of grid. For example, a simple first order equation is solved using different polynomial basis functions to illustrate the effectiveness of the algorithm. Then the theory of weighted Sobolev gradients is used for the singular Legendre's equation. Numerical experiments indicate that the new algorithm is more efficient than the previous algorithms discussed in the literature on the subject.

Approximating the solution of integro-differential problems via the spectral Tau method with filtering

Computing approximate solutions of integro-differential problems can be difficult and time consuming, requiring large expertize. Tau Toolbox is a numerical library that produces approximate polynomial solutions of differential, integral and integro-differential equations via the spectral Lanczos' Tau method. This approach, however, may fail to ensure the spectral rate of convergence, and even to reach convergence, whenever the exact solution shows singularities. This paper describes a post-processing phase based on a Frobenius-Padé approximation method to build rational approximations from the polynomial Tau approximation. This filtering extension improves the accuracy of the spectral approximation when working in the vicinity of solutions with singularities. Furthermore, some insights on how to perform such a filtering process on a piecewise version of the Tau method to tackle larger domains and/or stiff problems are offered.

Polynomial Least Squares Method for Fractional Lane–Emden Equations

This paper applies the Polynomial Least Squares Method (PLSM) to the case of fractional Lane-Emden differential equations. PLSM offers an analytical approximate polynomial solution in a straightforward way. A comparison with previously obtained results proves how accurate the method is.

Numerical solution of the first order nonlinear differen-tial equations with the mixed nonlinear conditions by using PLSM(comparison with Bernstein polynomials method)

We use the Polynomial Least Squares Method (PLSM), which allows us to compute analytical approximate polynomial solutions for nonlinear ordinary differential equations with the mixed nonlinear conditions. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using Bernstein polynomials method.

Polynomial approximate solutions of an unconfined Forchheimer groundwater flow equation

We consider a one-dimensional, unconfined groundwater flow equation for the horizontal propagation of water. This equation was derived by using a particular form of the Forchheimer equation in place of Darcy’s Law. Such equations can model turbulent flows in coarse and fractured porous media. For power-law head, exponential head, power-law flux and exponential flux boundary conditions at the inlet, the problems can be reduced, using similarity transformations, to boundary-value problems for a nonlinear ordinary differential equation. We construct quadratic and cubic approximate solutions of these problems. We also numerically compute solutions using a new modification of a method of Shampine, which exploits scaling properties of the governing equation. The polynomial approximate solutions replicate well the numerical solutions and they are easy to use. Last, we compare the predicted wetting front positions from our quadratic and cubic polynomials to predictions based on Adomian polynomials of the same degrees. The work demonstrates the value of polynomial approximate solutions for validating numerical solutions and for obtaining good approximations for water profiles and the extent of water propagation. This work also presents a new application of Shampine’s method for this type of groundwater flow equation. We note that this paper introduces additional classes of approximate solutions for the Forchheimer equation. Up to this date, not many solutions are known, especially for the transient cases considered here.

A numerical scheme to solve boundary value problems involving singular perturbation

In this study, a numerical method is presented in order to approximately solve singularly perturbed second-order differential equations givenwith boundary conditions. Themethod uses the set ofmonomialswhose degrees do not exceed a prescribed N as the set of base functions, resulting from the supposition that the approximate solution is a polynomial of degree N whose coefficients are to be determined. Then, following Galerkin’s approach, inner product with the base functions are applied to the residual of the approximate solution polynomial. This process, with a suitable incorporation of the boundary conditions, gives rise to an algebraic linear system of size N þ 1. The approximate polynomial solution is then obtained from the solution of this resulting system. Additionally, a technique, called residual correction,which exploits the linearity of the problem to estimate the error of any computed approximate solution is discussed briefly. The numerical scheme and the residual correction technique are illustrated with two examples.

A numerical method for solutions of Lotka–Volterra predator–prey model with time-delay

In this paper, we present a numerical scheme to obtain polynomial approximations for the solutions of continuous time-delayed population models for two interacting species. The method includes taking inner product of a set of monomials with a vector obtained from the problem under consideration. Doing this, the problem is transformed to a nonlinear system of algebraic equations. This system is then solved, yielding coefficients of the approximate polynomial solutions. In addition, the technique of residual correction, which aims to increase the accuracy of the approximate solution by estimating its error, is discussed in some detail. The method and the residual correction technique are illustrated with two examples.

An effective computational method for solving linear multi-point boundary value problems

In this work, an efficient computational method is proposed for solving the linear multi-point boundary value problems (MBVPs). Our approach depends mainly on of the least squares approximation method, the Lagrange-multiplier method and the residual error function technique. With the proposed scheme, we handle the numerical solutions of the linear MBVPs in a straightforward manner. Firstly, the given linear MBVP is reduced to a linear system of algebraic equations, and the coefficients of the approximate polynomial solution of the problem are determined by solving this system. Secondly, a linear boundary value problem related to the error function of the approximate solution is constructed, and error estimation is presented for the suggested method. The convergence of the approximate solution is proved. The reliability and efficiency of the proposed approach are demonstrated by some numerical examples.

Analytic Approximate Solutions for a Class of Variable Order Fractional Differential Equations Using The Polynomial Least Squares Method

Abstract In this paper a new way to compute analytic approximate polynomial solutions for a class of nonlinear variable order fractional differential equations is proposed, based on the Polynomial Least Squares Method (PLSM). In order to emphasize the accuracy and the efficiency of the method several examples are included.

Analytical approximate solutions for quadratic Riccati differential equation of fractional order using the Polynomial Least Squares Method

In this paper the Polynomial Least Squares Method is applied in order to compute analytical approximate polynomial solutions for the fractional Riccati type differential equations. The accuracy of the method is tested by means of the comparison with previous results for several applications.

A general polynomial solution to convection–dispersion equation using boundary layer theory

A number of models have been established to simulate the behaviour of solute transport due to chemical pollution, both in croplands and groundwater systems. An approximate polynomial solution to convection–dispersion equation (CDE) based on boundary layer theory has been verified for the use to describe solute transport in semi-infinite systems such as soil column. However, previous studies have only proposed low order polynomial solutions such as parabolic and cubic polynomials. This paper presents a general polynomial boundary layer solution to CDE. Comparison with exact solution suggests the prediction accuracy of the boundary layer solution varies with the order of polynomial expression and soil transport parameters. The results show that prediction accuracy increases with increasing order up to parabolic or cubic polynomial function and with no distinct relationship between accuracy and order for higher order polynomials (\(n\geqslant 3\)). Comparison of two critical solute transport parameters (i.e., dispersion coefficient and retardation factor), estimated by the boundary layer solution and obtained by CXTFIT curve-fitting, shows a good agreement. The study shows that the general solution can determine the appropriate orders of polynomials for approximate CDE solutions that best describe solute concentration profiles and optimal solute transport parameters. Furthermore, the general polynomial solution to CDE provides a simple approach to solute transport problems, a criterion for choosing the right orders of polynomials for soils with different transport parameters. It is also a potential approach for estimating solute transport parameters of soils in the field.

О сходимости полиномиальных приближенных решений уравнения минимальной поверхности

О сходимости полиномиальных приближенных решений уравнения минимальной поверхности

A New Technique for Solving Fractional Order Systems: Hermite Collocation Method

In this study, we establish an approximate method which produces an approximate Hermite polynomial solution to a system of fractional order differential equations with variable coefficients. At collocation points, this method converts the mentioned system into a matrix equation which corresponds to a system of linear equations with unknown Hermite polynomial coefficients. Construction of the method on the aforementioned type of equations has been presented and tested on some numerical examples. Results related to the effectiveness and reliability of the method have been illustrated.

The airfoil equation on near disjoint intervals: Approximate models and polynomial solutions

The airfoil equation is considered over two disjoint intervals. Assuming the distance between the intervals is small an approximate solution is found and relationships between this approximation and the solution of the classical airfoil equation are obtained. Numerical results show the convergence of the solution of the original problem to the approximation. Polynomial solutions for an approximate model are obtained and a spectral method for the generalized airfoil equation on near disjoint intervals is proposed.

Fast Computing on Vehicle Dynamics Using Chebyshev Series Expansions

This paper focuses on faster computation techniques to integrate mechanical models in electronic advanced active safety applications. It shows the different techniques of approximation in series of functions and differential equations applied to vehicle dynamics. This allows the achievement of approximate polynomial and rational solutions with a very fast and efficient computation. First, the whole theoretical basic principles related to the techniques used are presented: orthogonality of functions, function expansion in Chebyshev and Jacobi series, approximation through rational functions, the Minimax–Remez algorithm, orthogonal rational functions, and the perturbation of dynamic systems theory, that reduces the degree of the expansion polynomials used. As an application, it is shown the obtaining of approximate solutions to the longitudinal dynamics, vertical dynamics, steering geometry, and a tyre model, all obtained through development in series of orthogonal functions with a computation much faster than those of its equivalents in the classic vehicle theory. These polynomial partially symbolic solutions present very low errors and very favorable analytical properties due to their simplicity, becoming ideal for real-time computation as those required for the simulation of evasive manoeuvres prior to a crash. This set of techniques had never been applied to vehicle dynamics before.