A High-Order Iterative Scheme for a Nonlinear Pseudoparabolic Equation and Numerical Results

Nguyen Huu Nhan,Nguyen Thanh Long,Tran Trinh Manh Dung,Le Thi Phuong Ngoc,Le Thi Mai Thanh,Bekir Sahin

doi:10.1155/2021/8886184

Nguyen Huu Nhan, Nguyen Thanh Long + Show 4 more

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https://doi.org/10.1155/2021/8886184

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Abstract

In this paper, by applying the Faedo-Galerkin approximation method and using basic concepts of nonlinear analysis, we study the initial-boundary value problem for a nonlinear pseudoparabolic equation with Robin–Dirichlet conditions. It consists of two main parts. Part 1 is devoted to proof of the unique existence of a weak solution by establishing an approximate sequence u m based on a N -order iterative scheme in case of f ∈ C N 0,1 × 0 , T ∗ × ℝ N ≥ 2 , or a single-iterative scheme in case of f ∈ C 1 Ω ¯ × 0 , T ∗ × ℝ . In Part 2, we begin with the construction of a difference scheme to approximate u m of the N -order iterative scheme, with N = 2 . Next, we present numerical results in detail to show that the convergence rate of the 2-order iterative scheme is faster than that of the single-iterative scheme.

Highlights

In this paper, we consider the following initial-boundary problem: z t ut − 􏼠μ(t) +α(t) zt􏼡􏼒uxx x ux􏼓 􏽚 g(t − s) · 􏼒uxx(s)
We present numerical results in detail to show that the convergence rate of the 2-order iterative scheme is faster than that of the single-iterative scheme
In Part 1, by using the N-order iterative method, Faedo-Galerkin method, and compact method, we prove existence and uniqueness of a weak solution of problems (1)–(3)

Summary

Research Article

A High-Order Iterative Scheme for a Nonlinear Pseudoparabolic Equation and Numerical Results. By applying the Faedo-Galerkin approximation method and using basic concepts of nonlinear analysis, we study the initial-boundary value problem for a nonlinear pseudoparabolic equation with Robin–Dirichlet conditions. Part 1 is devoted to proof of the unique existence of a weak solution by establishing an approximate sequence 􏼈u(m)􏼉 based on a N-order iterative scheme in case of f ∈ CN([0, 1] × [0, T∗] × R)(N ≥ 2), or a single-iterative scheme in case of f ∈ C1(Ω × [0, T∗] × R). We present numerical results in detail to show that the convergence rate of the 2-order iterative scheme is faster than that of the single-iterative scheme

Introduction

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