Numerical Solution of Nonlinear Kaup-Kupershmit Equation, KdV-KdV and Hirota-Satsuma Systems

2012 ◽  
Vol 13 (7-8) ◽  
Author(s):  
Akbar Mohebbi
Keyword(s):  
Fourier Transform ◽  
Linear Operator ◽  
Differential Equations ◽  
Numerical Solution ◽  
Discrete Fourier Transform ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Time Stepping

AbstractIn this work we investigate the numerical solution of Kaup-Kupershmit (KK) equation, KdV-KdV and generalized Hirota-Satsuma (HS) systems. The proposed numerical schemes in this paper are based on fourth-order time-stepping schemes in combination with discrete Fourier transform. We discretize the original partial differential equations (PDEs) with discrete Fourier transform in space and obtain a system of ordinary differential equations (ODEs) in Fourier space which will be solved with fourth order time-stepping methods. After transforming the equations to a system of ODEs, the linear operator in KK and HS equation is diagonal but in KDV-KDV equation is not diagonal. However for KDV-KDV system which is the focus of this paper, we show that the exponential of linear operator and related inverse matrix have definite structure which enable us to implement the methods such as diagonal case. Comparing numerical solutions with exact traveling wave solutions demonstrates that those methods are accurate and readily implemented.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

Wavelet Galerkin method for fourth-order multi-dimensional elliptic partial differential equations

2018 ◽  
Vol 16 (05) ◽  
pp. 1850045 ◽  
Author(s):  
B. V. Rathish Kumar ◽  
Gopal Priyadarshi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Computational Cost ◽  
Stability Estimates ◽  
Partial Differential ◽  
Wavelet Galerkin Method

We describe a wavelet Galerkin method for numerical solutions of fourth-order linear and nonlinear partial differential equations (PDEs) in 2D and 3D based on the use of Daubechies compactly supported wavelets. Two-term connection coefficients have been used to compute higher-order derivatives accurately and economically. Localization and orthogonality properties of wavelets make the global matrix sparse. In particular, these properties reduce the computational cost significantly. Linear system of equations obtained from discretized equations have been solved using GMRES iterative solver. Quasi-linearization technique has been effectively used to handle nonlinear terms arising in nonlinear biharmonic equation. To reduce the computational cost of our method, we have proposed an efficient compression algorithm. Error and stability estimates have been derived. Accuracy of the proposed method is demonstrated through various examples.

Three-dimensional numerical modeling of gravity and magnetic anomaly in a mixed space-wavenumber domain

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. G41-G54 ◽  
Author(s):  
Shikun Dai ◽  
Dongdong Zhao ◽  
Shunguo Wang ◽  
Bin Xiong ◽  
Qianjiang Zhang ◽  
...  
Keyword(s):  
Numerical Modeling ◽  
Fourier Transform ◽  
Differential Equations ◽  
Numerical Solution ◽  
Magnetic Anomaly ◽  
Large Scale ◽  
Vertical Component ◽  
Wavenumber Domain ◽  
Gravity And Magnetic ◽  
Gravity And Magnetic Anomaly

Fast and accurate numerical modeling of gravity and magnetic anomalies is the basis of field-data inversion and quantitative interpretation. In gravity and magnetic prospecting, the computation and memory requirements of practical modeling is still a significant issue, which leads to the difficulty of using efficient and detailed inversions for large-scale complex models. A new 3D numerical modeling method for gravity and magnetic anomaly in a mixed space-wavenumber domain is proposed to mitigate the difficulties. By performing a 2D Fourier transform along two horizontal directions, 3D partial differential equations governing gravity and magnetic potentials in the spatial domain are transformed into a group of independent 1D differential equations wrapped with different wavenumbers. Importantly, the computation and memory requirements of modeling are greatly reduced by this method. A modeling example with 4,040,100 observations can be finished in approximately 28 s on a desktop using a single core, and the independent differential equations are highly parallel among different wavenumbers. The method preserves the vertical component in the space domain, and thus a mesh for modeling can be finer at a shallower depth and coarser at a deeper depth. In general, the new method takes into account the calculation accuracy and the efficiency. The finite-element algorithm combined with a chasing method is used to solve the transformed differential equations with different wavenumbers. In a synthetic test, a model with prism-shaped anomalies is used to verify the accuracy and efficiency of the proposed algorithm by comparing the analytical solution, our numerical solution, and a well-known numerical solution. Furthermore, we have studied the balance between computational accuracy and efficiency using a standard fast Fourier transform (FFT) method with grid expansion and the Gauss-FFT method. A model with topography is also used to explore the ability of modeling topography with our method. The results indicate that the proposed method using the Gauss-FFT method has characteristics of fast calculation speed and high accuracy.

NUMERICAL SOLUTION FOR THE NONLINEAR EMDEN–FOWLER TYPE EQUATIONS BY A FOURTH-ORDER ADAPTIVE METHOD

2013 ◽  
Vol 11 (01) ◽  
pp. 1350052 ◽  
Author(s):  
S. A. KHURI ◽  
A. SAYFY
Keyword(s):  
Finite Element ◽  
Numerical Solution ◽  
Rate Of Convergence ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Adaptive Method ◽  
B Splines ◽  
Special Cases ◽  
Collocation Approach

A finite element collocation approach, based on cubic B-splines, is manipulated for obtaining numerical solutions of a generalized form of the Emden–Fowler type equations. The rate of convergence is discussed theoretically and verified numerically to be of fourth-order by using the double-mesh principle. The efficiency of the scheme is tested on a number of examples which represent special cases of the problem under consideration. The results are compared with analytical and other numerical solutions that are available in the literature. The proposed method reveals that the outcomes are reliable and very accurate when contrasted with other existing methods.

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