Результаты поиска по 'partial solutions of inhomogeneous linear difference-differential equations':
Найдено статей: 2
  1. Tsirulik V.G.
    Calculation of particular solutions of nonhomogeneous linear equations with two linear operators, of which at least one is almost algebraic, in the case of simple roots of the characteristic equation
    Computer Research and Modeling, 2016, v. 8, no. 1, pp. 9-18

    The concept of an operator is an almost algebraic with respect to two-sided ideal of the algebra of linear operators in some finite-dimensional linear spaces, it extended to the case when the ideal is left. We prove a theorem on the following equation particular solution $\sum\limits^{n, m}_{i=0, j=0} a_{ij} A^i B^j u = f$, where $A$ and $B$ is a linear operator, $f$ is an element of a linear space. The result is applied to the differential-difference equations.

    Keywords: almost algebraic differential operators, almost algebraic difference operators, left regularizers of linear operators, differential-difference operators, partial solutions of inhomogeneous linear difference-differential equations.
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  2. Ainbinder R.M., Rassadin A.E.
    On population migration in an ecological niche with a spatially heterogeneous local capacity
    Computer Research and Modeling, 2025, v. 17, no. 3, pp. 483-500

    The article describes the migration process of a certain population, taking into account the spatial heterogeneity of the local capacity of the ecological niche. It is assumed that this spatial heterogeneity is caused by various natural or artificial factors. The mathematical model of the migration process under consideration is a Cauchy problem on a straight line for some quasi-linear partial differential equation of the first order, which is satisfied by the linear population density under consideration. In this paper, a general solution to this Cauchy problem is found for an arbitrary dependence of the local capacity of an ecological niche on the spatial coordinate. This general solution was applied to describe the migration of the population in question in two different cases: in the case of a dependence of the local capacity of the ecological niche on the spatial coordinate in the form of a smooth step and in the case of a hill-like dependence of the local capacity of the ecological niche on the spatial coordinate. In both cases, the solution to the Cauchy problem is expressed in terms of higher transcendental functions. By applying special relations to the model parameters, these higher transcendental functions are reduced to elementary functions, which makes it possible to obtain exact model solutions explicitly expressed in terms of elementary functions. With the help of these precise solutions, an extensive program of computational experiments has been implemented, showing how the initial population density of the Gaussian form is dispersed by the considered two types of spatial heterogeneity of the local capacity of the ecological niche. These computational experiments have shown that when passing through both step-like and hill-like spatial inhomogeneities of the local capacity of an ecological niche with a narrow Gaussian width of its initial density compared to the characteristic spatial scale of these inhomogeneities, the system forgets its initial state. In particular, if we interpret the system under study as a population living in an extended calm rectilinear river along its bed, then it can be argued that under this initial condition, after the current of this river carries the population under consideration through the area of spatial heterogeneity of the local capacity of the ecological niche, the population density becomes a quasi-rectangular function.

    Keywords: method of characteristics, Bernoulli equation, the Gaussian hypergeometric function, Appell hypergeometric function.

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