Stability of Traveling Waves to a Burgers Equation with 2nd-Order Nonlinear Diffusion

We are interested in the study of asymptotic stability for Burgers equation with second-order nonlinear diffusion. We first transform the original equation by the ansatz transformation to establish the existence of traveling wave. We further employ the energy estimate under small perturbation and arbitrary wave amplitude. This energy estimate is then used to establish the stability.

The Cauchy problem for inhomogeneous parabolic Shilov equations

In this paper, we consider the Cauchy problem for parabolic Shilov equations with continuous bounded coefficients. In these equations, the inhomogeneities are continuous exponentially decreasing functions, which have a certain degree of smoothness by the spatial variable. The properties of the fundamental solution of this problem are described without using the kind of equation. The corresponding volume potential, which is a partial solution of the original equation, is investigated. For this Cauchy problem the correct solvability in the class of generalized initial data, which are the Gelfand and Shilov distributions, is determined.

On solving non-homogeneous partial differential equations with right-hand side defined on the grid

An algorithm is proposed for obtaining solutions of partial differential equations with right-hand side defined on the grid $\{ x_{1}^{\mu}, x_{2}^{\mu}, \ldots, x_{n}^{\mu}\},\ (\mu=1,2,\ldots,N)\colon f_{\mu}=f(x_{1}^{\mu}, x_{2}^{\mu}, \ldots, x_{n}^{\mu}).$ Here $n$ is the number of independent variables in the original partial differential equation, $N$ is the number of rows in the grid for the right-hand side, $f=f( x_{1}, x_{2}, \ldots, x_{n})$ is the right-hand of the original equation. The algorithm implements a reduction of the original equation to a system of ordinary differential equations (ODE system) with initial conditions at each grid point and includes the following sequence of actions. We seek a solution to the original equation, depending on one independent variable. The original equation is associated with a certain system of relations containing arbitrary functions and including the partial differential equation of the first order. For an equation of the first order, an extended system of equations of characteristics is written. Adding to it the remaining relations containing arbitrary functions, and demanding that these relations be the first integrals of the extended system of equations of characteristics, we arrive at the desired ODE system, completing the reduction. The proposed algorithm allows at each grid point to find a solution of the original partial differential equation that satisfies the given initial and boundary conditions. The algorithm is used to obtain solutions of the Poisson equation and the equation of unsteady axisymmetric filtering at the points of the grid on which the right-hand sides of the corresponding equations are given.

Simplification method for nonlinear equations of monotone type in Banach space

In a Banach space, we study an operator equation with a monotone operator T. The operator is an operator from a Banach space to its conjugate, and T=AC, where A and C are operators of some classes. The considered problem belongs to the class of ill-posed problems. For this reason, an operator regularization method is proposed to solve it. This method is constructed using not the operator T of the original equation, but a more simple operator A, which is B-monotone, B=C−1. The existence of the operator B is assumed. In addition, when constructing the operator regularization method, we use a dual mapping with some gauge function. In this case, the operators of the equation and the right-hand side of the given equation are assumed to be perturbed. The requirements on the geometry of the Banach space and on the agreement conditions for the perturbation levels of the data and of the regularization parameter are established, which provide a strong convergence of the constructed approximations to some solution of the original equation. An example of a problem in Lebesgue space is given for which the proposed method is applicable.

MULTIPLICATIVE APPROXIMATION OF A RANDOM PROCESS

In this paper we consider the stochastic Ito differential equation in an infinite-dimensional real Hilbert space. Using the method of multiplicative representations of Daletsky - Trotter, its approximate solution is constructed. Under classical conditions on the coefficients, there is a single to the stochastic equivalence of solutions of the stochastic equation, which is a random process. This development generates an evolutionary family of resolving operators by the formula x(t)= S(t, Construct the division of the segment by the points. An equation with time-uniform coefficients is considered on each elementary segment . There is a single solution of this equation on the elementary segment, which generates the resolving operator by the formula The multiplicative expression is constructed. Using the method of Dalecki-Trotter multiplicative representations, it is proved that this multiplicative expression is stochastically equivalent to the representation generated by the solution of the original equation. This means that the specified multiplicative expression is respectively a representation of the solution of the original equation. That is, the probability of one coincides with the solution of the original stochastic equation. It should be noted that this is possible under additional conditions for the coefficients of the equation. These conditions are the time continuity of the coefficients of the equation. Thus, the constructed multiplicative representation can be interpreted as an approximate solution of the original equation. This method of multiplicative approximation makes it possible to simplify the study of the corresponding random process both at the elementary segment and as a whole. It is known, that the solution of a stochastic equation by a known formula generates a solution of the inverse Kolmogorov equation in the corresponding space. This scheme of multiplicative approximation can be transferred to the solution of the parabolic equation, which is the inverse Kolmogorov equation. Thus, the method of multiplicative approximation makes it possible to simplify the study of both stochastic equations and partial differential equations.

Contact Symmetries of a Model in Optimal Investment Theory

It is generally known that Lie symmetries of differential equations can lead to a reduction of the governing equation(s), lead to exact solutions of these equations and, in the best case scenario, lead to a linearization of the original equation. In this paper, we consider a model from optimal investment theory where we show the governing equation possesses an extensive contact symmetry and, through this, we show it is linearizable. Several exact solutions are provided including a solution to a particular terminal value problem.

Modifying an Equation to Predict the Asphaltene Deposition in the Buzurgan Oil Field

Buzurgan oil field suffers from the phenomenon of asphaltene precipitation. The serious negatives of this phenomenon are the decrease in production caused by clogging of the pores and decrease in permeability and wettability of the reservoir rocks, in addition to the blockages that occur in the pipeline transporting crude oil. The presence of laboratories in the Iraqi oil companies helped to conduct the necessary experiments, such as gas chromatography (GC) test to identify the components of crude oil and the percentages of each component, These laboratory results consider the main elements in deriving a new equation called modified colloidal instability index (MCII) equation based on a well-known global equation called colloidal instability index (CII) equation. The modified (MCII) equation is considered an equation compared to the original (CII) equation because both equations mainly depend on the components of the crude oil, but the difference between them lies in the fact that the original equation depends on the crude oil components at the surface conditions, while the new equation relies on the analysis of crude oil to its basic components at reservoir conditions by using (GC) analysis device. The components of the crude oil in the reservoir conditions according to the number of carbon atoms of each component compared with the elements of the original equation, which are (saturates, aromatics, resins, and asphaltene). The new MCII equation helps in predicting the possibility of asphaltene precipitation which can be used and generalized to other Iraqi oilfields as it has proven its worth and acceptability in this study.

On a linear autonomous descriptor equation with discrete time. II. Canonical representation and structural properties

We consider a linear homogeneous autonomous descriptor equation with discrete time B0g(k+1)+∑mi=1Big(k+1−i)=0,k=m,m+1,…, with rectangular (in general case) matrices Bi. Such an equation arises in the study of the most important control problems for systems with many commensurate delays in control: the 0-controllability problem, the synthesis problem of the feedback-type regulator, which provides calming to the solution of the original system, the modal controllability problem (controllability to the coefficients of characteristic quasipolynomial), the spectral reduction problem and the synthesis problem observers for dual surveillance system. The main method of the presented study is based on replacing the original equation with an equivalent equation in the “expanded” state space, which allows one to match the new equation of the beam of matrices. This made it possible to study a number of structural properties of the original equation by using the canonical form of the beam of matrices, and express the results in terms of minimal indices and elementary divisors. In the article, a criterion is obtained for the existence of a nontrivial admissible initial condition for the original equation, the verification of which is based on the calculation of the minimum indices and elementary divisors of the beam of matrices. The following problem was studied: it is required to construct a solution to the original equation in the form g(k+1)=Tψ(k+1), k=1,2…, where T is some matrix, the sequence of vectors ψ(k+1), k=1,2,…, satisfies the equation ψ(k+1)=Sψ(k), k=1,2,…, and the square matrix S has a predetermined spectrum (or part of the spectrum). The results obtained make it possible to construct solutions of the initial descriptor equation with predetermined asymptotic properties, for example, uniformly asymptotically stable.

Analytical and Numerical Solutions of the Riccati Equation Using the Method of Variation of Parameters. Application to Population Dynamics

This work presents new approximate analytical solutions for the Riccati equation (RE) resulting from the application of the method of variation of parameters. The original equation is solved using another RE explicitly dependent on the independent variable. The solutions obtained are easy to implement and highly applicable, which is confirmed by solving several examples corresponding to REs whose solution is known, as well as optimizing the method to determine the density of the members that make up a population. In this way, new perspectives are open in the study of the phenomenon of pattern formation.

Reduction and analytic solutions of a variable-coefficient Korteweg–de Vries equation in a fluid, crystal or plasma

In this paper, a variable-coefficient KdV equation in a fluid, plasma, anharmonic crystal, blood vessel, circulatory system, shallow-water tunnel, lake or relaxation inhomogeneous medium is discussed. We construct the reduction from the original equation to another variable-coefficient KdV equation, and then get the rational, periodic and mixed solutions of the original equation under certain constraint. For the original equation, we obtain that (i) the dispersive coefficient affects the solitonic background, velocity and amplitude; (ii) the perturbed coefficient affects the solitonic velocity, amplitude and background; (iii) the dissipative coefficient affects the solitonic background, and there are different mixed solutions under the same constraint with the dispersive, perturbed and dissipative coefficients changing.