Abstract
This paper investigates the one dimensional mixed problem with nonlocal boundary conditions, for the quasilinear parabolic equation. Under some natural regularity and consistency conditions on the input data, the existence, uniqueness, convergence of the weak generalized solution, and also continuous dependence upon the data of the solution are shown by using the generalized Fourier method. We construct an iteration algorithm for the numerical solution of this problem.
Highlights
D denotes the domainD := { < x
Denote the solution of problem ( )-( ) by u = u(x, t, ε). This problem was investigated with different boundary conditions by various researchers by using Fourier methods [ ]
We prove the existence, uniqueness convergence of the weak generalized solution, continuous dependent upon the data of the solution; and we construct an iteration algorithm for the numerical solution of this problem
Summary
Definition The function v(x, t) ∈ C (D ) is called test function if it satisfies the following conditions: v(x, T) = , v( , t) = v( , t), vx( , t) = , ∀t ∈ [ , T] and ∀x ∈ [ , ]. ∂ x dx = , for arbitrary test function v = v(x, t), is called a generalized (weak) solution of problem ( )-( ). We will use the Fourier series representation of the weak solution to transform the initial-boundary value problem to the infinite set of nonlinear integral equations. Of functions continuous on [ , T] satisfying the condition max. Theorem ( ) Assume the function f (x, t, u) is continuous with respect to all arguments in D × (–∞, ∞) and satisfies the following condition:.
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