scholarly journals A High Order Iterative Scheme for a Nonlinear Kirchhoff Wave Equation in the Unit Membrane

2011 ◽  
Vol 2011 ◽  
pp. 1-31
Author(s):  
Le Thi Phuong Ngoc ◽  
Nguyen Thanh Long
Keyword(s):  
Wave Equation ◽  
Weak Solution ◽  
Recent Result ◽  
Iterative Scheme ◽  
Convergent Sequence ◽  
High Order ◽  
Unique Weak Solution ◽  
Unit Membrane ◽  
Recurrent Sequence

A high-order iterative scheme is established in order to get a convergent sequence at a rate of orderN(N≥1) to a local unique weak solution of a nonlinear Kirchhoff wave equation in the unit membrane. This extends a recent result in (EJDE, 2005, No. 138) where a recurrent sequence converges at a rate of order 2.

scholarly journals High-Order Iterative Scheme for a Viscoelastic Wave Equation and Numerical Results

2021 ◽  
Vol 2021 ◽  
pp. 1-27
Author(s):  
Doan Thi Nhu Quynh ◽  
Bui Duc Nam ◽  
Le Thi Mai Thanh ◽  
Tran Trinh Manh Dung ◽  
Nguyen Huu Nhan
Keyword(s):  
Wave Equation ◽  
Numerical Solutions ◽  
Iterative Scheme ◽  
High Order ◽  
Recurrent Sequence ◽  
Viscoelastic Wave Equation ◽  
Proposed Model ◽  
Viscoelastic Wave ◽  
Convergent Rate ◽  
The Galerkin Method

In this paper, we consider a Robin problem for a viscoelastic wave equation. First, by the high-order iterative method coupled with the Galerkin method, the existence of a recurrent sequence via an N -order iterative scheme is established, and then the N -order convergent rate of the obtained sequence to the unique weak solution of the proposed model is also proved. Next, with N = 2 , a numerical algorithm given by the finite-difference method is constructed to approximate the solution via the 2-order iterative scheme. Moreover, the same algorithm for the single-iterative scheme generated by the 2-order iterative scheme is also considered. Finally, comparison with errors of the numerical solutions obtained by the single-iterative scheme and the 2-order iterative scheme shows that the convergent rate of the 2-order iterative scheme is faster than that of the single-iterative scheme.

scholarly journals Existence of a Unique Weak Solution to a Nonlinear Non-Autonomous Time-Fractional Wave Equation (of Distributed-Order)

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1283
Author(s):  
Karel Van Bockstal
Keyword(s):  
Wave Equation ◽  
Weak Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Order Differential Operator ◽  
Unique Weak Solution ◽  
Fractional Wave Equation ◽  
Equation Of Time ◽  
Initial Boundary Value ◽  
Distributed Order

We study an initial-boundary value problem for a fractional wave equation of time distributed-order with a nonlinear source term. The coefficients of the second order differential operator are dependent on the spatial and time variables. We show the existence of a unique weak solution to the problem under low regularity assumptions on the data, which includes weakly singular solutions in the class of admissible problems. A similar result holds true for the fractional wave equation with Caputo fractional derivative.

scholarly journals Existence of a Unique Solution to a Fractional Partial Differential Equation and Its Continuous Dependence on Parameters

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 851
Author(s):  
Robert Stegliński
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Weak Solution ◽  
Variational Methods ◽  
Unique Solution ◽  
Continuous Dependence ◽  
Fractional Laplacian ◽  
Unique Weak Solution ◽  
Fractional Partial Differential Equation ◽  
Integrodifferential Operator

In the present paper we give conditions under which there exists a unique weak solution for a nonlocal equation driven by the integrodifferential operator of fractional Laplacian type. We argue for the optimality of some assumptions. Some Lyapunov-type inequalities are given. We also study the continuous dependence of the solution on parameters. In proofs we use monotonicity and variational methods.

scholarly journals On a Nonlinear Wave Equation of Kirchhoff-Carrier Type: Linear Approximation and Asymptotic Expansion of Solution in a Small Parameter

2018 ◽  
Vol 2018 ◽  
pp. 1-18
Author(s):  
Nguyen Huu Nhan ◽  
Le Thi Phuong Ngoc ◽  
Nguyen Thanh Long
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Weak Solution ◽  
Dirichlet Problem ◽  
Small Parameter ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Existence And Uniqueness ◽  
Linearization Method ◽  
Asymptotic Expansion Of Solution

We consider the Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, the existence and uniqueness of a weak solution are proved. An asymptotic expansion of high order in a small parameter of a weak solution is also discussed.

On the construction and efficiency of staggered numerical differentiators for the wave equation

Geophysics ◽  
1990 ◽  
Vol 55 (1) ◽  
pp. 107-110 ◽  
Author(s):  
M. Kindelan ◽  
A. Kamel ◽  
P. Sguazzero
Keyword(s):  
Numerical Modeling ◽  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Computational Efficiency ◽  
Differential Operators ◽  
High Order

Finite‐difference (FD) techniques have established themselves as viable tools for the numerical modeling of wave propagation. The accuracy and the computational efficiency of numerical modeling can be enhanced by using high‐order spatial differential operators (Dablain,1986).

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