On the difference approximations of overdetermined hyperbolic equations of classical mathematical physics

Issues concerning difference approximations of overdetermined systems of hyperbolic equations are examined. The formulations of extended overdetermined systems are given for hydrodynamics equations, magnetohydrodynamics equations, Maxwell equations, and elasticity equations. Some approaches to the construction of difference schemes are discussed for these systems.     

An examination of the homentropic Euler equations with averaged characteristics

This paper examines the properties of the homentropic Euler equations when the characteristics of the equations have been spatially averaged. The new equations are referred to as the characteristically averaged homentropic Euler (CAHE) equations. An existence and uniqueness proof for the modified equations is given. The speed of shocks for the CAHE equations are determined. The Riemann problem is examined and a general form of the solutions is presented. Finally, numerically simulations on the homentropic Euler and CAHE equations are conducted and the behaviors of the two sets of equations are compared.     

Regularized shallow water equations and an efficient method for numerical simulation of shallow water flows

Regularized shallow water equations are derived as based on a regularization of the Navier-Stokes equations in the form of quasi-gasdynamic and quasi-hydrodynamic equations. Efficient finite-difference algorithms based on the regularized shallow water equations are proposed for the numerical simulation of shallow water flows. The capabilities of the model are examined by computing a test Riemann problem, the flow over an obstacle, and asymmetric dam break.     

Finding a complete set of solutions or proving unsolvability for certain classes of matrix polynomial equations with commuting coefficients

Two classes of matrix polynomial equations with commuting coefficients are examined. It is shown that the equations in one class have complete sets of solutions, whereas the equations in the other class are unsolvable. A method is given for finding the solution set of an equation in the former class.     

Conditions for partitioning a system of differential algebraic equations into weakly coupled subsystems

Systems of differential algebraic equations are examined. A method is proposed for transforming the rectangular matrix of algebraic equations to block diagonal form. This method ensures the prescribed accuracy of the solution with respect to the original system of equations.     

Characteristic functional equations of polynomials and the morera-carleman theorem

Several characteristic functional equations satisfied by classes of polynomials of bounded degree are examined in connection with certain generalizations of the Morera-Carleman Theorem. Certain functional equations which have nonanalytic polynomial solutions are also considered.     

Correct formulation and a spline method for solving a class of singular integro-differential equations

Conditions for the unique solvability of a class of singular integro-differential equations with Hilbertian kernels are examined. A spline method for solving such equations is justified, and numerical results are presented.     

Numerical solution of integral-algebraic equations for multistep methods

Systems of Volterra linear integral equations with identically singular matrices in the principal part (called integral-algebraic equations) are examined. Multistep methods for the numerical solution of a selected class of such systems are proposed and justified.     

Application of Quasihomogeneous Polynomials to Some Stability Problems

The quasihomogeneous polynomials examined previously by the author, especially quasiquadratic forms [1, 2], are used to study some stability problems. The stability of systems of differential equations with quasihomogeneous right-hand sides is studied. Theorems analogous to the first-approximation Lyapunov stability theorems are derived for one class of essentially nonlinear systems of a special form. Several problems for these systems are examined that may be among the critical cases for these nonlinear systems of differential equations.     

Stability and Conditional Stability of Time-dependent Difference Equations in Banach Spaces

Using methods parallel to those of ordinary differential equations,stability of time-dependent difference equations in Banach spacesis examined. Asymptotic stability, instability and conditionalstability properties are especially studied.     

Family of fifth-order three-level schemes for evolution problems

A multiparameter family of fifth-order three-level schemes in time based on compact approximations is presented for solving evolution problems. The schemes are adapted to hyperbolic and parabolic equations and to stiff systems of ordinary differential equations. In the case of hyperbolic equations, a fifth-order accurate scheme in all variables with compact approximations of spatial derivatives is analyzed. Stability estimates are presented, and the dispersive and dissipative properties are examined.     

On the solution of a class of integral equations II

A class of integral equations is investigated, particular examples of which occur in the consideration of certain three- and four-part mixed boundary-value problems in applied mathematics. A constructive method is given for reformulating the integral equations as Fredholm integral equations of the second kind and three examples are examined in detail to illustrate the general methods developed in the paper.     

On integrals of equations of unstable nearly self-similar flows : PMM vol. 39, n 6, 1975, pp. 1060–1067

One-dimensional or nearly one-dimensional unstable motions of perfect gas are considered. Integrals admitted by the system of equations defining such motions are examined. Since the existence of integrals is associated with some law of conservation, i. e. with some divergent form of presentation of equations of the input system, it is possible by examining all divergent equations of gasdynamics to derive certain new integrals not previously considered.     

Runge-Kutta collocation methods for differential-algebraic equations of indices 2 and 3

Stiffly accurate Runge-Kutta collocation methods with explicit first stage are examined. The parameters of these methods are chosen so as to minimize the errors in the solutions to differential-algebraic equations of indices 2 and 3. This construction results in methods for solving such equations that are superior to the available Runge-Kutta methods.     

Comparison of two approaches to integration on separated topological spaces

A class of integral equations is investigated, particular examples of which occur in the consideration of certain three- and four-part mixed boundary-value problems in applied mathematics. A constructive method is given for reformulating the integral equations as Fredholm integral equations of the second kind and three examples are examined in detail to illustrate the general methods developed in the paper.     

Application of semiorthogonal spline wavelets and the Galerkin method to the numerical simulation of thin wire antennas

The Bubnov-Galerkin method based on spline wavelets is used to solve singular integral equations. For the resulting systems of linear algebraic equations, the properties of their coefficient matrices are examined. Sparse approximations of these matrices are constructed by applying a cutting barrier. The results are used to numerically analyze thin wire antennas. Numerical results are presented.     

Symmetries, integrals and solutions of ordinary differential equations of maximal symmetry

Second- and third-order scalar ordinary differential equations of maximal symmetry in the traditional sense of point, respectively contact, symmetry are examined for the mappings they produce in solutions and fundamental first integrals. The properties of the ‘exceptional symmetries’, i.e. those not considered to be generic to scalar equations of maximal symmetry, can be recast into a form which is applicable to all such equations of maximal symmetry. Some properties of these symmetries are demonstrated.     

Scale dependent equations of motion of an ideal fluid

Starting from the property that the velocity is a divergence free filtered field we construct the equations of motion for an ideal fluid on a space–time-scale. Methods of approximation are briefly examined by which the macroscopic equations can be solved on individual scale slices. Validation of such approximations based on general residual models are discussed.     

On the integral equation formulations of some 2D contact problems

Two integral equations, representing the mechanical response of a 2D infinite plate supported along a line and subject to a transverse concentrated force, are examined. The kernels of the integral operators are of the type (x−y)ln|x−y| and (x−y)²ln|x−y|. In spite of the fact that these are only weakly singular, the two equations are studied in a more general framework, which allows us to consider also solutions having non-integrable endpoint singularities. The existence and uniqueness of solutions of the equations are discussed and their endpoint singularities detected.Since the two equations are of interest in their own right, some properties of the associated integral operators are examined in a scale of weighted Sobolev type spaces. Then, new results on the existence and uniqueness of integrable solutions of the equations that in some sense are complementary to those previously obtained are derived.     

Tikhonov solutions of approximately given systems of linear algebraic equations under finite perturbations of their matrices

The properties of a mathematical programming problem that arises in finding a stable (in the sense of Tikhonov) solution to a system of linear algebraic equations with an approximately given augmented coefficient matrix are examined. Conditions are obtained that determine whether this problem can be reduced to the minimization of a smoothing functional or to the minimal matrix correction of the underlying system of linear algebraic equations. A method for constructing (exact or approximately given) model systems of linear algebraic equations with known Tikhonov solutions is described. Sharp lower bounds are derived for the maximal error in the solution of an approximately given system of linear algebraic equations under finite perturbations of its coefficient matrix. Numerical examples are given.