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1

Comparison of modified ADM and classical finite difference method for some third-order and fifth-order KdV equations

Appadu Appanah Rao, Kelil Abey Sherif

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

377--409

EN

The KdV equation, which appears as an asymptotic model in physical systems ranging from water waves to plasma physics, has been studied. In this paper, we are concerned with dispersive nonlinear KdV equations by using two reliable methods: Shehu Adomian decomposition method (STADM) and the classical finite difference method for solving three numerical experiments. STADM is constructed by combining Shehu’s transform and Adomian decomposition method, and the nonlinear terms can be easily handled using Adomian’s polynomials. The Shehu transform is used to accelerate the convergence of the solution series in most cases and to overcome the deficiency that is mainly caused by unsatisfied conditions in other analytical techniques. We compare the approximate and numerical results with the exact solution for the two numerical experiments. The third numerical experiment does not have an exact solution and we compare profiles from the two methods vs the space domain at some values of time. This study provides us with information about which of the two methods are effective based on the numerical experiment chosen. Knowledge acquired will enable us to construct methods for other related partial differential equations such as stochastic Korteweg-de Vries (KdV), KdV-Burgers, and fractional KdV equations.

2

Deformed solitons of a typical set of (2+1)-dimensional complex modified Korteweg–de Vries equations

Yuan Feng, Zhu Xiaoming, Wang Yulei

International Journal of Applied Mathematics and Computer Science

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2020

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Vol. 30, no. 2

337--350

EN

Deformed soliton solutions are studied in a typical set of (2+1)-dimensional complex modified Korteweg–de Vries (cmKdV) equations. Through constructing the determinant form of the n-fold Darboux transformation for these (2+1)-dimensional cmKdV equations, we obtain general order-n deformed soliton solutions using zero seeds. With no loss of generality, we focus on order-1 and order-2 deformed solitons. Three types of order-1 deformed solitons, namely, the polynomial type, the trigonometric type, and the hyperbolic type, are derived. Meanwhile, their dynamical behaviors, including amplitude, velocity, direction, periodicity, and symmetry, are also investigated in detail. In particular, the formulas of |q[1]| and trajectories are provided analytically, which are involved by an arbitrary smooth function f(y + 4λ2t). For order-2 cases, we obtain the general analytical expressions of deformed solitons. Two typical solitons, possessing different properties in temporal symmetry, are discussed.

3

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

Yan J. L., Zheng L. H.

International Journal of Applied Mathematics and Computer Science

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2017

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Vol. 27, no. 3

515--525

EN

The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg–de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg–de Vries equation.

4

O(h5k) accurate finite difference method for the numerical solution of fourth order two point boundary value problems on geometric meshes

Jha N., Bieniasz L. K.

Czasopismo Techniczne. Nauki Podstawowe

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2016

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Y. 113, iss. 1-NP

55--72

EN

Two point boundary value problems for fourth order, nonlinear, singular and non-singular ordinary differential equations occur in various areas of science and technology. A compact, three point finite difference scheme for solving such problems on nonuniform geometric meshes is presented. The scheme achieves a fifth or sixth order of accuracy on geometric and uniform meshes, respectively. The proposed scheme describes the generalization of Numerov-type method of Chawla (IMA J Appl Math 24:35-42, 1979) developed for second order differential equations. The convergence of the scheme is proven using the mean value theorem, irreducibility, and monotone property of the block tridiagonal matrix arising for the scheme. Numerical tests confirm the accuracy, and demonstrate the reliability and efficiency of the scheme. Geometric meshes prove superior to uniform meshes, in the presence of boundary and interior layers.

PL

Dwupunktowe zagadnienia z warunkami brzegowymi, dla nieliniowych, osobliwych i nieosobliwych równań różniczkowych zwyczajnych czwartego rzędu, występują w różnych obszarach nauki i techniki. Zaprezentowano kompaktowy, trzypunktowy schemat różnicowy do rozwiązywania takich problemów na niejednorodnych siatkach geometrycznych. Schemat ten osiąga dokładność piątego lub szóstego rzędu, odpowiednio na siatkach geometrycznych lub jednorodnych. Proponowany schemat przedstawia uogólnienie metody typu Numerowa, autorstwa Chawli (IMA J Appl Math 24:35-42, 1979), opracowanej dla równań różniczkowych drugiego rzędu. Udowodniono zbieżność schematu, korzystając z twierdzenia o własności średniej, nieredukowalności oraz monotoniczności macierzy blokowo-trójdiagonalnej wynikającej ze schematu. Testy numeryczne potwierdzają dokładność, oraz demonstrują niezawodność i wydajność schematu. Siatki geometryczne wykazują przewagę nad siatkami jednorodnymi, w obecności warstw brzegowych i wewnętrznych.

5

Controllability of mixed Volterra–Fredholm type integrodifferential third order dispersion equations

Kumar M., Sukavanam N.

Journal of Applied Analysis

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2015

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Vol. 21, nr 1

1--7

EN

In this paper, we establish a result concerning the controllability of a mixed Volterra–Fredholm type integrodifferential third order dispersion equation. The result is obtained by using the theory of strongly continuous semigroups and the Banach fixed point theorem.