scholarly journals An inverse problem for a fourth-order Boussinesq equation with non-conjugate boundary and integral overdetermination conditions

2017 ◽  
pp. 17-36
Author(s):  
Yashar Megraliev ◽  
◽  
Farhad Alizade ◽  
Keyword(s):  
Inverse Problem ◽  
Fourth Order ◽  
Boussinesq Equation ◽  
Integral Overdetermination

scholarly journals Regarding New Wave Patterns of the Newly Extended Nonlinear (2+1)-Dimensional Boussinesq Equation with Fourth Order

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 341 ◽  
Author(s):  
Juan Luis García Guirao ◽  
Haci Mehmet Baskonus ◽  
Ajay Kumar
Keyword(s):  
Fourth Order ◽  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Bright Soliton ◽  
New Wave ◽  
Wave Patterns

This paper applies the sine-Gordon expansion method to the extended nonlinear (2+1)-dimensional Boussinesq equation. Many new dark, complex and mixed dark-bright soliton solutions of the governing model are derived. Moreover, for better understanding of the results, 2D, 3D and contour graphs under the strain conditions and the suitable values of parameters are also plotted.

scholarly journals A Fourth Order Finite Difference Method for the Good Boussinesq Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
M. S. Ismail ◽  
Farida Mosally
Keyword(s):  
Exact Solution ◽  
Nonlinear System ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Fourth Order ◽  
Boussinesq Equation ◽  
First Order ◽  
System A ◽  
Accuracy And Stability ◽  
Linearization Techniques

The “good” Boussinesq equation is transformed into a first order differential system. A fourth order finite difference scheme is derived for this system. The resulting scheme is analyzed for accuracy and stability. Newton’s method and linearization techniques are used to solve the resulting nonlinear system. The exact solution and the conserved quantity are used to assess the accuracy and the efficiency of the derived method. Head-on and overtaking interactions of two solitons are also considered. The numerical results reveal the good performance of the derived method.

On the inverse problem of calculus of variations for fourth-order equations

2010 ◽  
Vol 466 (2120) ◽  
pp. 2309-2323 ◽  
Author(s):  
M. C. Nucci ◽  
A. M. Arthurs
Keyword(s):  
Inverse Problem ◽  
Number Theory ◽  
Calculus Of Variations ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Order Equation ◽  
Fourth Order Equation ◽  
Link Type ◽  
Even Order ◽  
Fourth Order Equations

Fels’ conditions (Fels, M. E. 1996 Trans. Amer. Math. Soc. 348 , 5007–5029. ( doi:10.1090/S0002-9947-96-01720-5 )) ensure the existence and uniqueness of the Lagrangian in the case of a fourth-order equation. We show that when Fels’ conditions are satisfied, the Lagrangian can be derived from the Jacobi last multiplier, as in the case of a second-order equation. Indeed, we prove that if a Lagrangian exists for an equation of any even order, then it can be derived from the Jacobi last multiplier. Two equations from a Number Theory paper by Hall (Hall, R. R. 2002 J. Number Theory 93 , 235–245. ( doi:10.1006/jnth.2001.2719 )), one of the second and one of the fourth order, will be used to exemplify the method. The known link between Jacobi last multiplier and Lie symmetries is also exploited. Finally, the Lagrangians of two fourth-order equations drawn from Physics are determined with the same method.

scholarly journals Determination of a time-dependent heat source under not strengthened regular boundary and integral overdetermination conditions

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 809-814 ◽  
Author(s):  
Makhmud Sadybekov ◽  
Gulaiym Oralsyn ◽  
Mansur Ismailov
Keyword(s):  
Inverse Problem ◽  
Heat Source ◽  
Input Data ◽  
Nonlocal Boundary ◽  
Fourier Method ◽  
Time Dependent ◽  
Regular Boundary ◽  
Associated Functions ◽  
Integral Overdetermination

We investigate an inverse problem of finding a time-dependent heat source in a parabolic equation with nonlocal boundary and integral overdetermination conditions. The boundary conditions of this problem are regular but not strengthened regular. The principal difference of this problem is: the system of eigenfunctions is not complete. But the system of eigen- and associated functions forming a basis. Under some natural regularity and consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the solution are shown by using the generalized Fourier method.

High Order Energy-Preserving Method of the “Good” Boussinesq Equation

2016 ◽  
Vol 9 (1) ◽  
pp. 111-122 ◽  
Author(s):  
Chaolong Jiang ◽  
Jianqiang Sun ◽  
Xunfeng He ◽  
Lanlan Zhou
Keyword(s):  
Vector Field ◽  
Fourth Order ◽  
Boussinesq Equation ◽  
Field Method ◽  
High Order ◽  
Discrete Energy ◽  
High Order Scheme ◽  
Order Energy ◽  
Order Scheme ◽  
Pseudo Spectral Method

AbstractThe fourth order average vector field (AVF) method is applied to solve the “Good” Boussinesq equation. The semi-discrete system of the “good” Boussinesq equation obtained by the pseudo-spectral method in spatial variable, which is a classical finite dimensional Hamiltonian system, is discretizated by the fourth order average vector field method. Thus, a new high order energy conservation scheme of the “good” Boussinesq equation is obtained. Numerical experiments confirm that the new high order scheme can preserve the discrete energy of the “good” Boussinesq equation exactly and simulate evolution of different solitary waves well.

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