Abstract
Two-dimensional parabolic equation with nonlocal condition is solved by alternating direction method in the rectangular domain. Values of the solution on the boundary points are bind with the integral of the solution in whole two-dimensional domain. Because of this nonlocal condition, the classical alternating direction method is complemented by the solution of low dimension system of algebraic equations. The peculiarities of the method are considered.
Highlights
Introduction and statement of the problemThe particularity of our problem (1)–(5) under consideration is that the value of the solution in nonlocal condition (4) at the boundary points is linked with a two-dimensional integral of the solution
Introduction and statement of the problemThe paper deals with the initial problem for parabolic equation with an integral boundary condition∂u ∂2u ∂2u ∂t = ∂x2 + ∂y2 + f (x, y, t), x, y ∈ Ω = {0 < x, y < 1}, 0 < t T, (1)u(0, y, t) = μ1(y, t), u(1, y, t) = μ2(y, t), (2)u(x, 1, t) = μ3(x, t), (3)u(x, 0, t) = γ(x) u(ξ, η, t) dξdη + μ4(x, t), x ∈ Γ1, (4)Ω u(x, y, 0) = φ(x, y). (5)
The particularity of our problem (1)–(5) under consideration is that the value of the solution in nonlocal condition (4) at the boundary points is linked with a two-dimensional integral of the solution
Summary
The particularity of our problem (1)–(5) under consideration is that the value of the solution in nonlocal condition (4) at the boundary points is linked with a two-dimensional integral of the solution. This is the difference of our research from the analogous researches in the above mentioned articles [2,3,4,5,6,7,8,9,10]. The main result of our paper is the fact that we show that two-dimensional parabolic equations with a nonlocal condition of type (4) can be successfully solved by an efficient alternating direction method, and that condition (7) is not always necessary for this purpose
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