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The effect of undulated bottom on the shallow water waves is studied on the assumption that change in depth is sufficiently small and its characteristic length is much larger than the wave length. For small amplitude waves, an eigenvalue equation is obtained to give the conditions for appearance of trapped mode. The system of equations governing finite amplitude waves is derived. It is reduced to the modified Korteweg-de Vries equation with two additional terms. The stability of soliton for two dimensional perturbation is also studied.

The critical load of structures sensitive to initial imperfections may be considerably less then the classical buckling load. A method to evaluate the critical load of reticulated shells with rigid joint. based on Koiter's general post-buckling theory is introduced. The use of generalized strain-displacement relationships and vector algebra leads to a polynomial representation of the static, buckling and post-buckling equation. As a result, the numerical solution is simplified and consists of solution of systems of linear equations and an eigenvalue problem. The slope of the post-buckling curve is obtained by the aid of which the critical load is computed.

The linearized equations governing the deformations of incompressible elastic bodies are discussed. The Dirichlet problem is formulated for this system of equations using the theory of elliptic systems due to Douglis and Nirenberg. A uniqueness theorem is proved. Necessary and sufficient conditions for uniqueness of solution to the Dirichlet problem are obtained for small deformations of a Mooney-Rivlin material which has been subjected to a finite homogeneous biaxial deformation.

Ordinary iteration schemes for solving linear algebraic equations are one-step, i.e., only the last iterant is used in order to compute the following one. This note advocates the use of several iterants by means of least-square acceleration. The resulting scheme is easy to implement and is very effective in cases where the basic iteration matrix is close to symmetric. The note provides theoretical estimates for the rate of convergence as well as a numerical example. The example deals with the numerical solution of Poisson's equation in a rectangular annulus, by five-point formulae, using the alternating-direction method.

For pt.I, see abstr. A7075 of 1973. The coupled integro-differential equations, described in Paper I, are reduced to linear algebraic equations. The finite difference formulae used are described in detail. The linear algebraic equations are solved using a technique which has good numerical stability and which gives good linear independence in the solution vectors. The method is economic in the use of computer time and storage.

Starting with the three-dimensional theory of classical continuum mechanics, some aspects of both the nonlinear and the linear theories of elastic rods are discussed. Detailed attention is given to the derivation of constitutive equations for the linear isothermal theory of elastic rods of an isotropic material and of variable cross-section, deduced by an approximation procedure from the three-dimensional equations. Explicit linear constitutive relations are obtained for straight isotropic circular rods of non-uniform cross-section; the explicit calculation is carried out (in terms of an approximate specific Gibbs free energy function) in four distinct parts, since the complete system of equations involved separate into those appropriate for extensional, torsional and two flexural modes of deformation. A system of displacement differential equations is derived for flexure of a beam of variable circular cross-section; they reduce to those of the Timoshenko beam theory when the radius of the cross-section is constant.

Thus we have proposed a method of processing of experts' estimates that is based on the use of formulas (3)–(4) and requires for the determination of the “weights” a preliminary interrogation of the experts for the purpose of finding the elements of the matrix A={aij}. The elements of this matrix must be subjected to the normalization conditions (8) and also to the conditions (7) or to the stronger conditions (12) or (13). After the interrogation it is necessary to solve a system of linear equations (9), (6) that always has a solution ci, with ci≥0 for i=1,..., n. If the matrix A is indecomposable [in particular, if the conditions (12) or (13) hold], this solution will be unique and strictly positive, i.e., ci≥0 for i=1,..., n. During the solution it is possible to drop one of the equations of system (9) (by virtue of the linear dependence of these equations), and regard the remaining n-1 equations of (9) together with Eq. (6) as a linear algebraic system of n equations with n unknowns ci.

An algorithm for computing exactly a general solution to a system of linear equations with coefficients that are polynomials over the integers is presented. The algorithm applies mod-p mappings and then evaluation mappings, eventually solving linear systems of equations with coefficients in GF(p) by a special Gaussian elimination algorithm. Then by applying interpolation and the Chinese Remainder Theorem a general solution is obtained. For a consistent system, the evaluation-interpolation part of the algorithm computes the determinantal RRE form of the mod-p reduced augmented system matrices. The Chinese Remainder Theorem then uses these to construct an RRE matrix with polynomial entries over the integers, from which a general solution is constructed. For an inconsistent system, only one mod-p mapping is needed. The average computing time for the algorithm is obtained and compared to that for the exact division method. The new method is found to be far superior. Also, a mod-p/evaluation mapping algorithm for computing matrix products is discussed briefly.

The dynamic contact problem for a half-plane reinforced on its boundary by a finite elastic strip is considered. The solution of the problem reduces to solving an integral equation of the first kind, and then an infinite system of linear equations by using Chebyshev polynomials. It is proved that this infinite system of equations is quasi-completely regular. Moreover, a simple analytical expression, completely admissible for practical applications and differing by an arbitrarily small amount from the exact expression, is obtained for the kernel of the integral equation. In this case, for definite values of some physical parameter, the complete regularity of the appropriate infinite system of equations is proved in addition to the quasi-complete regularity and numerical results are obtained for the law of variation of the amplitude of the tangential contact stresses under the strip. The problem under consideration is related to problems of load transfer from stringers to elastic solids which are important for engineering practice. The case of an infinite or semi-infinite strip has been examined earlier [1].

The method of simplifying systems of equations with small nonlinearities and dispersion is considered. Such systems differ from the linear hyperbolic system by a certain integro-differential operator with a small parameter. Method is based on the reduction of input equations to the normal form and subsequent recurrent procedure. In the case of a wave propagating along one of the characteristics of the system (single-wave processes) the first approximation by this method leads to known Burgers, Korteweg-de Vries, Klein-Gordon, and others equations which were first derived for specific physical models, and later for a more general system of nonlinear equations [1, 2], However the proposed method can be applied to multi-wave interaction, when waves corresponding to two or more characteristics of the system are taken into consideration. It is also shown that in the case of single-, as well as multi-wave processes the order of simplified equations increases with the order of approximation, and the increase is different for systems with high and low dispersion frequencies (when the small parameter is contained in terms with higher derivatives or those containing integrals of the unknown function, respectively).

A principle is introduced to the thermodynamics of irreversible processes which generalizes the well-known Onsager principle. On the basis of this principle, an equation of heat conduction is derived with a finite velocity of heat propagation, and a system of equations of coupled thermoelasticity is set up.

A medium which may be slightly inhomogeneous in space and time is characterized by its current response. The author derives a system of equations describing the linear and non-linear coupling of WKB modes.

A method for the solution of large systems of linear equations is presented, based on a tridiagonal partition with varying order of the submatrices. Though formally similar to the Potters' algorithm it removes all of its significant limitations.

The first part of this paper deals with the Frey-Salmon irreversibility hypothesis, the kinetic equation that is deduced therefrom, and the comparison with experiment for the viscosity and thermal conductivity coefficients. The relaxation time which has been introduced is the average value of the duration of a collision. The second part deals with a new irreversibility hypothesis which leads to a kinetic equation of the same form as the first but in which the relaxation time is deduced from the interparticle potential.

A system of equations for predicting the evaporation rates of solvents is presented. These equations may be used by the industrial hygienist, in conjunction with the diffusion equations developed elsewhere, to predict the atmospheric concentrations of vapors from spilled toxic liquids. The equations are derived from wind tunnel tests and applied to predictions for spills both indoors and outdoors.

A comprehensive theory of the matrix linear equation $AX + XB = C$ is presented. The equation is viewed as a vector equation $LX = C$ in the vector space of all $m \times n$ matrices. As the main result, two necessary and sufficient conditions for a solution to exist and a general form of all solutions are established.

The numerical solution of systems of differential equations of the formB dx/dt=σ(t)Ax(t)+f(t),x(0) given, whereB andA (withB and —(A+AT) positive definite) are supposed to be large sparse matrices, is considered.A-stable methods like the Implicit Runge-Kutta methods based on Radau quadrature are combined with iterative methods for the solution of the algebraic systems of equations.

The governing equations of the steady state creep of a two-dimensional thin shallow circular cylindrical shell are developed on the basis of Mises' criterion and the power law of creep. Stresses, membrane forces and bending moments expressed in terms of displacements are then linearized by expanding them in the neighbourhood of a certain approximate value of displacement. Substitution of these expressions into the equations of equilibrium reduces the problem into a set of simultaneous linear differential equations with respect to the small perturbation of the displacements. A method that may be interpreted as a modification of the Newton-Raphson method combined with the method of finite differences is used to solve the linear system of equations. Although calculations are made for a cylindrical panel with clamped edges subjected to normal pressure, the method is quite general and other types of shells, boundary conditions and geometries can be treated similarly.

Contrary to the common assumption, it is shown that the general return difference and the general null return difference are variant under the general transformations between a system of basis circuits and a system of basis cutsets. Necessary and sufficient conditions are presented for the invariance of these functions. Formulation of these functions in terms of the primary systems of equations and illustrative examples are also given.

Let x=A(t)x be a system of two linear ordinary differential equations with almost periodic coefficients. Then there exists for any positive e an almost reducible system of equations x=B(t)x with almost periodic coefficients and such that sup ∥A(t)−B (t)∥

Singular perturbation methods are used to determine the field diffracted by a semi-infinite rigid plate of thickness 2 a under irradiation by a plane acoustic wave at wavenumber k . Six terms of both an outer and an inner expansion in the small parameter ε = ka are calculated in closed form, yielding simple results for the far-field directivity pattern. The outer series is determined by certain eigensolutions, and by a sequence of straightforward Wiener-Hopf problems, while the inner terms are all obtained from a simple conformal mapping. Previous discussions of this problem (e.g. Jones 1953) involve the formulation of a modified Wiener-Hopf equation, and reduce the problem to that of inverting an infinite matrix with elements dependent upon ε. Jones has given a numerical inversion of the truncated 4 x 4 matrix in the limit ε→ 0. Here we obtain exact expressions A 2n+1 = 1 / 2(2n+1) { J n (n+½) - J n+1 (n+½)} for Jones’s variables, and prove that they satisfy his infinite system of linear equations. It is also shown that to O (ε 2 In 2 ε) the plate may be replaced by a duct longer than the plate by an amount L = ( a /π) In 2, in agreement with Jones’s numerical value of L = 0.22 a , together with the monopole field necessary to annul the effect of plane wave propagation down the duct. The principle used here to match the inner and outer asymptotic expansions is a slightly, though significantly, modified version of the one used extensively by Van Dyke (1964). In the present problem Van Dyke’s matching principle appears to hold, in that matching can be formally accomplished, but leads to erroneous results violating the reciprocal theorem . Accordingly, an appendix here gives a discussion and justification, in elementary terms, of the proposed modified asymptotic matching principle.

The problem of computing a desired function value to within a prescribed tolerance can be formulated in the following two distinct ways: Formulation I: Given x and ∈ > 0, compute f (x) to within ∈. Formulation II: Given only that x is in a closed interval X, compute a subinterval of the image, f (X) = { f (x) : x ∈ X}. The first formulation is applicable when x is known to arbitrary accuracy. The second formulation is applicable when x is known only to a limited accuracy, in which case the tolerance is prescribed albeit indirectly by the interval X, and one must be satisfied with all or part of the set f (X) of possible function values. Elsewhere the author has presented an efficient solution to Formulation I for any rational f and many nonrational f . B. A. Chartres has presented an efficient solution to Formulation II for a very restricted class of rational f and for a few nonrational f . In this paper a solution to Formulation II for the arbitrary nonconstant rational f is presented. By bounding df/dx away from zero over some subset of X, it is shown how to reduce Formulation II to Formulation I, yielding the solution given here. In generalizing to vector-valued functions f, Chartres has solved Formulation II only for rational f which satisfy a linear system of equations, while this paper presents a solution for arbitrary non-degenerate rational vector-valued f.

In a multi-server queueing system in which the customers are of several different types, it is useful to define states which specify the types of customers being served as well as the total number present. Analogies with some problems in statistical mechanics are found fruitful. Certain generating functions are defined in such a way that they satisfy a system of linear equations. Solution of the associated eigenvector problem shows that the steady-state probabilities for states in which all the servers are busy can be represented by a weighted sum of geometric probabilities.

Previous attempts to identify aircraft stability and control derivatives from flight test data, using three-degrees-of-freedom (3-DOF) longitudinal or lateral-directional perturbation equation-of-motion models, suffer from the disadvantage that the coupling between the longitudinal and lateral-directional dynamics has been ignored. In this paper, the identification of aircraft stability parameters is accomplished using a more accurate 6-DOF model which includes this coupling. Hierarchical system identification theory is used to reduce the computational effort involved. The 6-DOF system of equations is first decomposed into two 3-DOF subsystems, one for the longitudinal dynamics and the other for the lateral-directional dynamics. The two subsystem parameter identification processes are then coordinated in such a way that the overall system parameter identification problem is solved. Next, a six-subsystem decomposition is considered. Computational considerations and comparison with the unhierarchically structured problem are presented.

The calculation of the flow rate, power requirement and thermal energy of the extrusion machine requires preliminary knowledge of the expressions for the velocity and temperature profiles along the depth and for the pressure variation along the length of the screw channel The system of 19 equations with 19 unknowns which describes the thermomechanical process in the transition zone cannot, in general, be integrated directly owing to the complex interdependence between the velocity, temperature and pressure profiles and the dependence on temperature and pressure of the physical characteristics of the material which is processed. The paper presents a method for solving the system of equations which describes the thermomechanical process occurring in the screw channel and a procedure by means of which the expressions for the velocity, temperature and pressure profiles can be obtained. The adopted method is such that these expressions also take into account the heat developed by internal friction. It is assumed that the thermomechanical process in the machine is non-isothermal and polytropic and that the pressure at a point of the screw channel is anistropic.

This paper presents a necessary and sufficient condition for the existence of solutions to a system of mixed linear equations. This condition is expressed in terms of generalized inverses and one obtains the general form for all solutions when the system is consistent.

A connection is established between the production of entropy in irreversible processes and the Lyapunov function of the corresponding system of equations. Thermodynamic limitations are formulated on the functions that enter in the kinetic equations. The results are illustrated with relaxation processes in viscoelastic media.

The system of linear equations of multicomponent convective diffusion is reduced consistently with allowance for the features of the relevant laser devices. A criterion for the realization of a purely diffusion regime is established. A method of diagonalization of the diffusion coefficient matrix is given; it reduces the multicomponent problem to a series of one-component problems. The superposition of relaxation modes is discussed. A diffusion type hydrofluoric laser illustrates the influence of angular asymmetry of the particles on the output power of the radiation.

A combinational circuit realizing the switching function f(x) may be regarded as a solution verifier for the Boolean equation f(x) = 1. (*) The output of the circuit is 1, that is, if and only if the input-vector x is a solution for (*). We use the term "equational logic" to denote an approach to circuit synthesis based on (*) rather than on the function f(x). The central problem of equational logic is to find a system of equations gi(x) = hi(x) (i = 1,2,...,k), of the simplest possible form, that has the same solutions as (*). Given such a k-equation system, f(x) may be realized as the output of a k-wide digital comparator whose inputs are the 2k g's and h's constituting the system. The problem of finding a simple system of equations equivalent to a given equation was investigated more than a century ago by Willian Stanley Jevons, who called it "the inverse problem of logic." It was thought by Jevons and other 19th century logicians that the inverse problem is "always tentative," i.e., that it does not admit of algorithmic solution. It is shown in this paper, however, that the inverse problem may be solved as a covering problem by use of the "table of consequences" of Poretsky. As presently formulated, this approach is limited in practical utility by the large size of the tables involved. It appears that a practical solution technique requires a reformulation of the inverse problem.

A method for finding a first integral of the motion of a system of equations written in Hamiltonian form, for the case in which no “classical integrals” exist, is introduced and derived in this paper. The present method is based on a canonical transformation applied to the state vector of the system with the objective of obtaining a new Hamiltonian that is time-separable in the new state vector.

A numerical generalized-capacity-matrix technique is developed for application to aerodynamic flow computations. This technique allows the very fast direct (noniterative) numerical elliptic solvers to be used in poblems with arbitrary internal boundaries and with a wide class of boundary conditions, including numerical application of the Kutta condition on an airfoil without iteration. Accuracy, speed, and usefulness of the technique are demonstrated with linear problems for potential flows over airfoil shapes. The method's main advantages, however, can be exploited within iterative procedures for a variety of complex flow problems governed by systems of equations not necessarily elliptic or linear

The known Bethe and Cowley methods are extended to the case of ternary alloys with body centered cubic (BCC) lattice and three kinds of sites. The system of equations to determine the Curie point of an A2BC type alloy is obtained.

A new method for network analysis, the “force density method”, is presented. This concept is based upon the force-length ratios or force densities which are defined for each branch of the net structure. It is shown that the force densities are very suitable parameters for the description of the equilibrium state of any general network: the associated node coordinates of the structure are obtained by solving one single system of linear equations. A Gaussian transformation of the topological branch-node matrix yields the equation matrix. A method based on force densities renders a simple linear “analytical form finding” possible. If the free choice of the force densities is restricted by further conditions, the linear method is extended to a nonlinear one. A damped least squares approach is then introduced. In this version the method may be applied to the detailed computation of networks. Moreover, an interesting cross-connection between geometrical minimum way nets and prestressed unloaded structures is derived using the force densities.

A description is given of the set of algebraic equations that connect the components of the solutions of a system of linear differential equations. The result thus obtained is applied to the proof of a theorem on the estimates of the orders of the zeros.

A very efficient computer subroutine for the direct solution of large numbers of simultaneous linear equations is presented. Basically the program uses Gauss elimination on positive-definite symmetrical systems. The specific features are that systems of very large size and bandwidth can be solved and that all operations on zero elements are eliminated. Also, the program is very simple and can be incorporated into existing programs with a minimum of effort. The amount of backup storage available on the computer used will govern the maximum size of the system which can be solved. A FORTRAN IV listing of the subroutine is given.

This paper describes a fast and numerically stable method for solving the discrete Dirichlet problem for Poisson's equation in case of a rectangle (and mainly, a square). By using a special calculus for difference operators, the system of linear equations is reduced to a block-triangular system such that the diagonal blocks are heavily diagonally dominant. For a standard version of the algorithm, the number of operations and the computing time are proportional toh ?2 (h=mesh width). The method is one oftotal reduction compared with the method ofblock-cyclic reduction (odd-even reduction) [2], which we describe as a method ofpartial reduction.--Due to the developed calculus, many generalizations are possible.--In a following part II of the paper, the algorithm and numerical results will be described in detail.

Let A be a real symmetric positive definite n × n matrix and b a real column n-veetor. The problem of finding n-vectors x and y such that Ax = b + y, xTy = 0, x ≥ 0 and y ≥ 0, occurs when the method of Christopherson is used to solve free boundary problems for journal bearings. In this case A is a ‘ finite difference ’ matrix. We present a direct method for solving the above problem by solving a number of linear systems Akx = bk. Each system can be solved using one of the recently developed fast direct or iterative procedures. We consider the solution by factorization techniques.

The method examined here for modeling large combination units and complexes, using a method of decomposition according to process channels in combination with a two-model representation of the channel model, permits a considerable reduction in dimensionality of the optimization problem and the development of a system of equations operating in real time.

A finite-difference method for the integration of the linear two-point boundary value problem $$y'' = f(x)y + g(x), y(a) = y_a , y(b) = y_b $$ is constructed. It uses values off′,f″,g′,g″ at the grid points to obtainO(h8) global error, and allows strict global error bounds, while needing only the solution of a tridiagonal system of equations. Numerical examples are presented to demonstrate the practical usefulness of our method.

An analytical method is developed to solve approximately the bifurcation buckling load of a circular cylindrical shell under nonaxisymmetrical lateral pressure. The assumption of semi-inextensible deformation is made, which sets the circumferential strain to zero and simplifies the system of equations as well as the boundary conditions. In order to investigate the accuracy of the new method, solutions are first obtained for a cylindrical shell under uniform lateral pressure with either simple-simple or clamped-free boundary conditions. Comparisons are made in the former case to Flugge’s theory and, in the latter case, to a finite element analysis. Agreement in both cases is very good. The theory is then applied to find the buckling load of a clamped-free cylindrical shell under nonaxisymmetrical lateral pressure created by wind. Disagreement with a previous theory by Langhaar and Miller is examined. Comparisons with results of some experiments are also made and possible reasons of the discrepancies are considered.