scholarly journals Construction of Analytic Solution for Time-Fractional Boussinesq Equation Using Iterative Method

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Fei Xu ◽  
Yixian Gao ◽  
Weipeng Zhang
Keyword(s):  
Analytical Solution ◽  
Iterative Method ◽  
Fourth Order ◽  
Iterative Process ◽  
Boussinesq Equation ◽  
Analytic Solution ◽  
Boussinesq Equations ◽  
Sixth Order ◽  
Linear And Nonlinear

This paper is aimed at constructing analytical solution for both linear and nonlinear time-fractional Boussinesq equations by an iterative method. By the iterative process, we can obtain the analytic solution of the fourth-order time-fractional Boussinesq equation inR,R2, andRn, the sixth-order time-fractional Boussinesq equation, and the2nth-order time-fractional Boussinesq equation inR. Through these examples, it shows that the method is simple and effective.

scholarly journals Explicit solutions of generalized nonlinear Boussinesq equations

2001 ◽  
Vol 1 (1) ◽  
pp. 29-37 ◽  
Author(s):  
Doğan Kaya
Keyword(s):  
Power Series ◽  
Boussinesq Equation ◽  
Analytic Solution ◽  
Boussinesq Equations ◽  
The Self ◽  
Explicit Solutions ◽  
Convergent Power Series ◽  
Decomposition Scheme ◽  
Adomian Decomposition ◽  
Generalized Boussinesq Equation

By considering the Adomian decomposition scheme, we solve a generalized Boussinesq equation. The method does not need linearization or weak nonlinearly assumptions. By using this scheme, the solutions are calculated in the form of a convergent power series with easily computable components. The decomposition series analytic solution of the problem is quickly obtained by observing the existence of the self-canceling “noise” terms where sum of components vanishes in the limit.

scholarly journals Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jinxing Liu ◽  
Xiongrui Wang ◽  
Jun Zhou ◽  
Huan Zhang
Keyword(s):  
Lower Bounds ◽  
Fourth Order ◽  
Boussinesq Equation ◽  
Blow Up ◽  
Sixth Order ◽  
Upper And Lower Bounds ◽  
Differential Inequalities ◽  
Nonlinear Source ◽  
Order Dispersion ◽  
Dispersion Term

<p style='text-indent:20px;'>This paper deals with the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. By using some ordinary differential inequalities, the conditions on finite time blow-up of solutions are given with suitable assumptions on initial values. Moreover, the upper and lower bounds of the blow-up time are also investigated.</p>

scholarly journals Application of double natural decomposition method for solving singular one dimensional Boussinesq equation

Filomat ◽  
2018 ◽  
Vol 32 (12) ◽  
pp. 4389-4401 ◽  
Author(s):  
Hassan Gadain
Keyword(s):  
Decomposition Method ◽  
Boussinesq Equation ◽  
Adomian Decomposition Method ◽  
Boussinesq Equations ◽  
Science And Engineering ◽  
Transform Method ◽  
One Dimensional ◽  
Adomian Decomposition ◽  
Linear And Nonlinear ◽  
Natural Decomposition

In this work, a combined form of the double Natural transform method with the Adomian decomposition method is developed for analytic treatment of the linear and nonlinear singular one dimensional Boussinesq equations. Two examples are provided to illustrate the simplicity and reliability of this method. Moreover, the results show that the proposed method is effective and is easy to implement for certain problems in science and engineering.

A Blended Compact Difference (BCD) Method for Solving 3D Convection–Diffusion Problems with Variable Coefficients

2019 ◽  
Vol 17 (06) ◽  
pp. 1950022 ◽  
Author(s):  
Tingfu Ma ◽  
Yongbin Ge
Keyword(s):  
Fourth Order ◽  
Numerical Solutions ◽  
Sixth Order ◽  
Test Problems ◽  
Convection Diffusion ◽  
Compact Difference ◽  
Second Derivatives ◽  
Order Scheme ◽  
Linear And Nonlinear ◽  
Grid Points

In this study, we present a fourth-order and a sixth-order blended compact difference (BCD) schemes for approximating the three-dimensional (3D) convection–diffusion equation with variable convective coefficients. The proposed schemes, where transport variable, its first and second derivatives are carried as the unknowns, combine virtues of compact discretization, fourth-order Padé scheme and sixth-order combined compact difference (CCD) scheme for spatial derivatives and can efficiently capture numerical solutions of linear and nonlinear convection–diffusion equations with Dirichlet boundary conditions. The fourth-order scheme requires only 7 grid points and the sixth-order scheme requires 19 grid points. The distinguishing feature of the present method is that methodologies of explicit compact difference and implicit compact difference are blended together. The truncation errors of the two difference schemes are analyzed for the interior grid points, respectively. Simultaneously, a sixth-order accuracy scheme is proposed to compute the first and second derivatives of the grid points on boundaries. Finally, the presented methods are applied to several test problems from the literature including linear and nonlinear problems. It is found that the presented schemes exhibit good performance.

An Efficient Sixth-Order Convergent Newton-Type Iterative Method for Nonlinear Equations

2012 ◽  
Vol 220-223 ◽  
pp. 2585-2588
Author(s):  
Zhong Yong Hu ◽  
Fang Liang ◽  
Lian Zhong Li ◽  
Rui Chen
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Efficiency Index ◽  
Sixth Order ◽  
Type Method ◽  
Second Derivatives ◽  
Better Than

In this paper, we present a modified sixth order convergent Newton-type method for solving nonlinear equations. It is free from second derivatives, and requires three evaluations of the functions and two evaluations of derivatives per iteration. Hence the efficiency index of the presented method is 1.43097 which is better than that of classical Newton’s method 1.41421. Several results are given to illustrate the advantage and efficiency the algorithm.

scholarly journals Two solutions to Kirchhoff-type fourth-order implusive elastic beam equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jian Liu ◽  
Wenguang Yu
Keyword(s):  
Iterative Method ◽  
Variational Methods ◽  
Critical Point ◽  
Fourth Order ◽  
Elastic Beam ◽  
Kirchhoff Type ◽  
Newton Iterative Method ◽  
Beam Equations ◽  
Critical Point Theorems ◽  
Elastic Beam Equations

AbstractIn this paper, the existence of two solutions for superlinear fourth-order impulsive elastic beam equations is obtained. We get two theorems via variational methods and corresponding two-critical-point theorems. Combining with the Newton-iterative method, an example is presented to illustrate the value of the obtained theorems.

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