Abstract
The asymptotic method is a very attractive area of applied mathematics. There are many modern research directions which use a small parameter such as statistical mechanics, chemical reaction theory and so on. The application of the Fokker-Planck equation (FPE) with a small parameter is the most popular because this equation is the parabolic partial differential equations and the solutions of FPE give the probability density function. In this paper we investigate the singularly perturbed Cauchy problem for symmetric linear system of parabolic partial differential equations with a small parameter. We assume that this system is the Tikhonov non-homogeneous system with constant coefficients. The paper aims to consider this Cauchy problem, apply the asymptotic method and construct expansions of the solutions in the form of two-type decomposition. This decomposition has regular and border-layer parts. The main result of this paper is a justification of an asymptotic expansion for the solutions of this Cauchy problem. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations.
Highlights
It is well known that the differential operator, which is applied in the theory of measure, has such form:L = aij∂xi ∂xj + bi∂xi, 1 ⩽ i, j ⩽ d, d ∈ N .The solution of the equation L∗μ = 0 is Borel measures on an open set Ω ∈ Rd and there is the relation∫ Lfdμ = 0, ∀f ∈ C0∞(Ω)
We use the asymptotic method for this Cauchy problem and construct expansions of solutions in the form of decomposition, which has regular and border-layer parts
Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker–Planck equations
Summary
It is well known that the differential operator, which is applied in the theory of measure, has such form:. We can formulate the singularly perturbed Cauchy problem for FPE in the form: ε∂tρ(x, t, ε) − ∂xi ∂xj aij(x, t)ρ(x, t, ε) + ∂xi bi(x, t)ρ(x, t, ε) = 0, ρ(x, 0, ε) = ρ0(x), x ∈ Ω, ∀ρ0(x) ∈ C0∞(Ω), where ε > 0 is a small parameter. Perjan [22] obtains the asymptotic expansions of the solutions to the Cauchy problem for the linear symmetric hyperbolic system as the small parameter ε → 0. In this paper we apply the results of the paper [21] and investigate the Cauchy problem for the singularly perturbed Tikhonov-type symmetric system of non-homogeneous constant coefficients linear parabolic partial differential equations (LPPDE system) with a small parameter. We use the asymptotic method for this Cauchy problem and construct expansions of solutions in the form of decomposition, which has regular and border-layer parts. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker–Planck equations
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