Modification of the Euler Polygonal Method for Solving a Semi-periodic Boundary Value Problem for Pseudo-parabolic Equat
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Modification of the Euler Polygonal Method for Solving a Semi-periodic Boundary Value Problem for Pseudo-parabolic Equation of Special Type A. T. Assanova
and S. S. Kabdrakhova
Abstract. A semi-periodic boundary value problem for a pseudoparabolic equation of special type is investigated. A modification of the Euler polygonal method is applied to a semi-periodic boundaryvalue problem for a non-classical differential equation of third order. By introducing new unknown functions, the problem under consideration is reduced to an equivalent problem consisting of a family of periodic boundary value problems for a system of two ordinary differential equations and some integral relations. We obtain the conditions for the unique solvability of the problem. The estimates of convergence of approximate solution of the equivalent problem to the exact solution of the original problem are established. Mathematics Subject Classification. 34A45, 34B08, 34K13, 35G15, 35S15. Keywords. Pseudo-parabolic equation, semi-periodic boundary value problem, family of periodic boundary value problems, modification of the Euler polygonal method.
1. Introduction On the domain Ω = [0, ω] × [0, T ], we consider the semi-periodic boundary value problem for the non-classical partial differential equation of third order ∂2u ∂u ∂3u + a1 (x, t) = a (x, t) 0 ∂x∂t2 ∂x∂t ∂x ∂2u ∂u + a4 (x, t)u + f (x, t), +a2 (x, t) 2 + a3 (x, t) ∂t ∂t u(0, t) = ψ(t), t ∈ [0, T ],
(x, t) ∈ Ω, (1.1) (1.2)
This investigation is supported by Grant of the Ministry Education and Science of the Republic of Kazakhstan, no. AP 05131220.
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A. T. Assanova and S. S. Kabdrakhova
u(x, 0) = u(x, T ), x ∈ [0, ω], ∂u(x, 0) ∂u(x, T ) = , x ∈ [0, ω], ∂t ∂t
MJOM
(1.3) (1.4)
where the function u(x, t) is unknown, ai (x, t), i = 0, 4, and f (x, t) are continuous on Ω, ψ(t) is twice continuously differentiable on [0, T ], and the com˙ ˙ ) hold. patibility conditions ψ(0) = ψ(T ), ψ(0) = ψ(T Let C(Ω, R) and C([0, T ], R) be the spaces of continuous functions u : Ω → R and ψ : [0, T ] → R, respectively. A function u(x, t) is called a solution to problems (1.1)–(1.4) if it has ∂u(x, t) ∂u(x, t) ∂ 2 u(x, t) ∂ 2 u(x, t) , , , continuous on Ω partial derivatives , ∂x ∂t ∂t2 ∂x∂t 3 ∂ u(x, t) and satisfies Eq. (1.1) and conditions (1.2)–(1.4). ∂x∂t2 Boundary value problems for partial differential equations of third order describe real processes of mechanics, nonlinear acoustics, and magnetic fluid dynamics. Longitudinal vibrations of composite rods consisting of elastic and elastic-viscous sections are modeled by partial differential equations of third order [1,3,5,6,8,12,14,17–19,21,22,26–33,35–42]. There is a close relationship between third-order partial differential equations and biological phenomena [1–7,34]. For example, the propagation of nerve impulses is described by the FitzHugh–Nagumo model interconnected with the perturbed sine-Gordon equation, which is a semi-linear version of the third-order equation (1.1). These equations arise in the study of J
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