Efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields

Finite Difference Solution to Thermoelastic Field in a Thin Circular FGM Disk with a Concentric Hole

A new technique for solving the large system of linear algebraic equations associated with implicit differencing of multidimensio nal partial differential equations is presented. The coefficient matrix of the equations is factored, and then approximations to certain terms in the matrix are obtained from series expansions. The resulting system of equations is solved easily. The method is developed and demonstrated using a simple representative two-dimensional equation. Very good results are obtained when one direction is dominant. MPLICIT finite-difference schemes for the solution of multidimensional partial differential equations are usually stable and therefore applicable to a large class of problems. However, they are difficult to implement and may require an excessive amount of computer storage and time. The long computing time arises from the need to solve the large system of linear algebraic equations that result from the differencing. The computing time can be reduced significantly by approximating the coefficient matrix of the linear equations with a matrix that produces a system of equations that are relatively easy to solve. Among such methods are the alternating direction method (ADI)l used by Beam and Warming2 and Stone's strongly implicit method,3 which has been tested by Linetal.4 In this paper, a new technique for solving the large system of linear algebraic equations associated with implict differencing of multidimensional partial differential equations is presented. This method, called the pseudo-elimination method (PE), is shown to be faster than Stone's method for certain problems. The method is directly applicable to linear and linearized nonlinear systems of parabolic or elliptic partial differential equations. In order to discuss the method, a simple linear partial differential equation will be used; however, it should be kept in mind that the PE method is applicable to much more complicated problems. The question of whether the method will work when applied to difficult problems is not addressed. The scope of this paper is limited to presenting the method and illustrating, via a simple problem, that the method has some merit and deserves further study.

This paper studies rational and Liouvillian first integrals of homogeneous linear differential systems Y′=AY over a differential field k. Following [26], our strategy to compute them is to compute the Darboux polynomials associated with the system. We show how to explicitly interpret the coefficients of the Darboux polynomials as functions on the solutions of the system; this provides a correspondence between Darboux polynomials and semi-invariants of the differential Galois groups, which in turn gives indications regarding the possible degrees for Darboux polynomials (particularly in the completely reducible cases). The algorithm is implemented and we give some examples of computations.KeywordsLinear Differential EquationGalois GroupDifferential AlgebraCyclic VectorSymmetric PowerThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Applications of the Lanczos method

We present a new parallel algorithm for computing a least squares solution to a sparse overdetermined system of linear equations Ax=b such that m×n matrix A is sparse and the graph, G=(V,E), of the matrix \(H = \left[ {\begin{array}{*{20}c}I & {A^T } \\A & O \\\end{array} } \right]\) has an s(m+n)-separator family, that is, either |V| <n0 for a fixed constant n0 or, by deleting a separator subset S of vertices of the size ≤ s(m+n), G can be partitioned into two disconnected subgraphs having vertex sets V1,V2 of the sizes ≤ 2/3 (m+n) and each of the two resulting subgraphs induced by the vertex sets S ∪ Vi, i=1,2, can be recursively s(|S ∪ Vi|)-separated in a similar way. Our algorithm uses O(log (m+n) log2s(m+n)) steps and ≤ s3(m+n) processors; it relies on our recent parallel algorithm for solving sparse linear systems and has several immediate applications of interest, in particular to mathematical programming, to sparse nonsymmetric systems of linear equations, and to the path algebra computations. We most closely examine the impact on the linear programming problem, l.p.p., which requires to maximize cTy subject to ATy ≤ b, y ≥ 0 where A is an m×n matrix. Hereafter it is assumed that m ≥ n. The recent algorithm by N. Karmarkar gives the best known upper estimate (O(m3.5 L) arithmetic operations where L is the input size) for the cost of the solution of this problem in the worst case. We prove an asymptotic improvement of that result in the case where the graph of the associated matrix H has an s(m+n)-separator family; then our algorithm can be implemented using O(m L log m log2s(m+n)) parallel arithmetic steps, s3(m+n) processors and a total of O(m L s3(m+n) log m log2s(m+n)) arithmetic operations. In many cases of practical importance this is a considerable improvement of the known estimates: for example, s(m+n) = √8 (m+n) if G is planar (as occurs in many operations research applications, for instance, in the problem of computing the maximum multicommodity flow with a bounded number of commodities in a network having an s(m+n)-separator family), so that the processor bound is only 8 √8 (m+n)1.5 and the total number of arithmetic steps is O(m2.5L) in that case. Similarly Karmarkar's algorithm and the known algorithms for the solution of overdetermined linear systems are accelerated in the case of dense input matrices via our recent parallel algorithms for the inversion of dense k×k matrices using O(log2k) steps, k3 processors. Combined with a modification of Karmarkar's algorithm, this implies solution of l.p.p. using O(Lm log2m) steps, m2.5 processors. The stated results promise some important practical applications. Theoretically the above processor bounds can be reduced for dense matrix inversion to o(k2.5) and for l.p.p. to o(m2.165) in the dense case and to o(s2.5(m+n)) in the sparse case (preserving the same number of parallel steps); this also decreases the sequential time bound for the l.p.p. by a factor of m0.335, that is, to O(Lm3.165).

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

This paper focuses on the derivation, analysis and implementation of a hybrid method by optimizing the order of the method by introduction of six-hybrid points for direct solution of fifth order ordinary differential equations of initial value problems (IVPs). Power series was used as the basis function for the solution of the IVP. The basis function was interpolated at some selected hybrid points whereas the fifth derivative of the approximate solution was collocated at all the interval of integration of the method to generate a system of linear equations for the determination of the unknown parameters. The derived method was tested for consistency, zero stability, convergence and absolute stability. The method was tested with two linear test problems to confirm its accuracy and usability. The comparison of the results with some existing methods shows the superiority of the accuracy of the method.

Structural stability of planar quasi-homogeneous vector fields

Numerical solution for high order differential equations using a hybrid neural network—Optimization method

A new order m – 1 rational integrator with an inhomogeneuous constant

An efficient algorithm based on group theory and the Woodbury formula for the dynamic responses of periodic structures

A system of linear interval equations is called solvable if each system of linear equations contained therein is solvable. In the main result of this paper it is proved that solvability of a general rectangular system of linear interval equations can be characterized in terms of nonnegative solvability of a finite number of systems of linear equations which, however, is exponential in matrix size; the problem is proved to be NP-hard. It is shown that three earlier published results are consequences of the main theorem, which is compared with its counterpart valid for linear interval inequalities that turn out to be much less difficult to solve.

At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.

The present work deals with numerical solution of advection-dispersion equation (ADE) to find solute distribution profiles along and against groundwater flow in two-dimensional finite homogeneous porous medium. Initially, the transport medium is supposed as non-solute free. A constant concentration is assigned throughout the medium at the initial time. An intermediate point is located in the medium from where dispersion of solute is studied along and against groundwater flow. Dirichlet boundary conditions are prescribed along co-ordinate axes from the intermediate point. Solute fluxes are prescribed as zeros at the extreme boundaries. For the numerical solution, alternating direction implicit (ADI) method is applied to approximate the ADE together with initial and boundary conditions. On approximation of the ADE by ADI method, a tridiagonal linear system of algebraic equations is obtained at each half level of time interval. This linear system of equations is solved graphically to illustrate solute transport.

Darboux polynomials and rational first integrals of the generalized Lorenz systems