On the solvability and numerical methods for solution of linear integro-algebraic equations

We study solvability conditions for a system of Volterra equations with some identically degenerate or rectangular matrix at the main term. Connection is discussed of the solvability conditions and applicability of numerical methods for solving these systems. In particular, the conditions of the convergence of the least squares method with the error functional defined in Sobolev spaces are presented.     

The numerical solution of linear ordinary differential equations by feedforward neural networks

It is demonstrated, through theory and examples, how it is possible to construct directly and noniteratively a feedforward neural network to approximate arbitrary linear ordinary differential equations. The method, using the hard limit transfer function, is linear in storage and processing time, and the L₂ norm of the network approximation error decreases quadratically with the increasing number of hidden layer neurons. The construction requires imposing certain constraints on the values of the input, bias, and output weights, and the attribution of certain roles to each of these parameters.

All results presented used the hard limit transfer function. However, the noniterative approach should also be applicable to the use of hyperbolic tangents, sigmoids, and radial basis functions.     

On the numerical solution of linear and nonlinear volterra integral and integro-differential equations

Sinc bases are developed to approximate the solutions of linear and nonlinear Volterra integral and integro-differential equations. Properties of these sinc bases and some operational matrices are first presented. These properties are then used to reduce the integral and integro-differential equations to systems of linear or nonlinear algebraic equations. Numerical examples illustrate the pertinent features of the method and its applicability to a large variety of problems. The examples include convolution type, singular as well as singularly-perturbed problems.     

A reformulation of Olver's algorithm for the numerical solution of second-order linear difference equations

Summary A reformulation of an algorithm developed by F.W.J. Olver for the numerical solution of second-order difference equations is presented. It requires less computations and needs no adaptations in case certain coefficients vanish. It has also the advantage of avoiding overflow in some special cases.     

Finding numerical solution of linear partial differential equations of first order with real coefficients

Applying Bittner's operational calculus we present a method to give approximate solutions of linear partial differential equations of first order

Analytical and numerical methods for the stability analysis of linear fractional delay differential equations

In this paper, several analytical and numerical approaches are presented for the stability analysis of linear fractional-order delay differential equations. The main focus of interest is asymptotic stability, but bounded-input bounded-output (BIBO) stability is also discussed. The applicability of the Laplace transform method for stability analysis is first investigated, jointly with the corresponding characteristic equation, which is broadly used in BIBO stability analysis. Moreover, it is shown that a different characteristic equation, involving the one-parameter Mittag-Leffler function, may be obtained using the well-known method of steps, which provides a necessary condition for asymptotic stability. Stability criteria based on the Argument Principle are also obtained. The stability regions obtained using the two methods are evaluated numerically and comparison results are presented. Several key problems are highlighted.     

Wall-cooling effect on linear stability of compressible streamwise corner-flow

The influence of constant wall temperatures on the compressible flow along a right-angled corner and its stability behaviour is investigated by temporal local linear stability analysis. Wall-cooling up to 90 % of the adiabatic wall-temperature for a subsonic flow (Ma = 0.95) is considered. The maximum cross-flow velocity along the corner bi-sector for the base flow with adiabatic wall boundary condition is higher than for the base flow with constant adabatic wall-temperature. For increasing wall temperature, the amplification rate of the viscous modes decreases to a lesser extent than that of the corner mode, therefore the critical Reynolds-number is further on defined by the first viscous mode. The spatial behaviour is displayed using N-factors computed from Gaster-transformed modes. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Taylor method for numerical solution of generalized pantograph equations with linear functional argument

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this paper, we introduce a numerical method based on the Taylor polynomials for the approximate solution of the pantograph equation with retarded case or advanced case. The method is illustrated by studying the initial value problems. The results obtained are compared by the known results.     

Strong solution to the compressible magnetohydrodynamic equations with vacuum

We consider the compressible magnetohydrodynamic (MHD) equations with nonnegative thermal conductivity or infinite electric conductivity. We prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set.     

Nonnegative integral solution of linear equations

A method to obtain a nonnegative integral solution of a system of linear equations, if such a solution exists is given. The method writes linear equations as an integer programming problem and then solves the problem using a combination of artificial basis technique and a method of integer forms.     

The numerical solution of linear recurrence relations

By analysing the effect of rounding error on linear recurrence relations having exponential-type complementary solutions, the distribution of two-point boundary conditions most likely to specify a well-conditioned problem is deduced. An intuitive explanation is derived by extending the Bernouilli method of polynomial factorisation to the case of variable coefficients. The replacement of specific boundary conditions by a knowledge of the relative behaviour of the particular and complementary solutions is discussed, and illustrated by examples.     

A collocation approach for the numerical solution of certain linear retarded and advanced integrodifferential equations with linear functional arguments

This article presents a Taylor collocation method for the approximate solution of high‐order linear Volterra‐Fredholm integrodifferential equations with linear functional arguments. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. Some numerical examples, which consist of initial and boundary conditions, are given to show the properties of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Accurate solution of polynomial equations using Macaulay resultant matrices

We propose an algorithm for solving two polynomial equations in two variables. Our algorithm is based on the Macaulay resultant approach combined with new techniques, including randomization, to make the algorithm accurate in the presence of roundoff error. The ultimate computation is the solution of a generalized eigenvalue problem via the QZ method. We analyze the error due to roundoff of the method, showing that with high probability the roots are computed accurately, assuming that the input data (that is, the two polynomials) are well conditioned. Our analysis requires a novel combination of algebraic and numerical techniques.

     

Uniqueness of integer solution of linear equations

We consider the system of m linear equations in n integer variables Ax = d and give sufficient conditions for the uniqueness of its integer solution x ∈ {−1, 1} ⁿ by reformulating the problem as a linear program. Necessary and sufficient uniqueness characterizations of ordinary linear programming solutions are utilized to obtain sufficient uniqueness conditions such as the intersection of the kernel of A and the dual cone of a diagonal matrix of ±1’s is the origin in R ⁿ . This generalizes the well known condition that ker(A) = 0 for the uniqueness of a non-integer solution x of Ax = d. A zero maximum of a single linear program ensures the uniqueness of a given integer solution of a linear equation.