An efficient method for fractional nonlinear differential equations by quasi-Newton's method and simplified reproducing kernel method

2017 ◽  
Vol 41 (1) ◽  
pp. 5-14 ◽  
Author(s):  
Minqiang Xu ◽  
Jing Niu ◽  
Yingzhen Lin
Keyword(s):  
Differential Equations ◽  
Efficient Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Nonlinear Differential Equations ◽  
Reproducing Kernel Method

scholarly journals A New Approach for Solving Nonlinear Singular Boundary Value Problems

2018 ◽  
Vol 23 (1) ◽  
pp. 33-43
Author(s):  
Hui Zhu ◽  
Jing Niu ◽  
Ruimin Zhang ◽  
Yingzhen Lin
Keyword(s):  
Boundary Value Problems ◽  
Newton’S Method ◽  
Newton's Method ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Boundary Value ◽  
Singular Boundary ◽  
Singular Boundary Value Problems ◽  
Reproducing Kernel Method ◽  
New Approach

In this paper, an efficient method based on Quasi-Newton's method and the simpliffied reproducing kernel method is proposed for solving nonlinear singular boundary value problems. For the Quasi-Newton's method the convergence order is studied. The uniform convergence of the numerical solution as well as its derivatives are also proved. Numerical examples are given to demonstrate the efficiency and stability of the proposed method. The numerical results are compared with exact solutions and the outcomes of other existing numerical methods.

Marine Pipeline Analysis Based on Newton’s Method With an Arctic Application

1970 ◽  
Vol 92 (4) ◽  
pp. 827-833 ◽  
Author(s):  
D. W. Dareing ◽  
R. F. Neathery
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Newton’S Method ◽  
Finite Differences ◽  
Newton's Method ◽  
Nonlinear Differential Equations ◽  
Bending Moments ◽  
Iterative Solutions ◽  
Linear Differential ◽  
Pipe Deflection

Newton’s method is used to solve the nonlinear differential equations of bending for marine pipelines suspended between a lay-barge and the ocean floor. Newton’s method leads to linear differential equations, which are expressed in terms of finite differences and solved numerically. The success of Newton’s method depends on initial trial solutions, which in this paper are catenaries. Iterative solutions converge rapidly toward the exact solution (pipe deflection) even though large bending moments exist in the pipe. Example calculations are given for a 48-in. pipeline suspended in 300 ft of water.

Numerical simulation of time-fractional partial differential equations arising in fluid flows via reproducing Kernel method

2019 ◽  
Vol 30 (11) ◽  
pp. 4711-4733 ◽  
Author(s):  
Omar Abu Arqub
Keyword(s):  
Numerical Simulation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Multiphase Flows ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Kernel Method ◽  
Fractional Partial Differential Equations ◽  
Reproducing Kernel Method ◽  
Content Type

Purpose The subject of the fractional calculus theory has gained considerable popularity and importance due to their attractive applications in widespread fields of physics and engineering. The purpose of this paper is to present results on the numerical simulation for time-fractional partial differential equations arising in transonic multiphase flows, which are described by the Tricomi and the Keldysh equations of Robin functions types. Design/methodology/approach Those resulting mathematical models are solved by using the reproducing kernel method, which provide appropriate solutions in term of infinite series formula. Convergence analysis, error estimations and error bounds under some hypotheses, which provide the theoretical basis of the proposed method are also discussed. Findings The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the prospects of the gained results and the method are discussed through academic validations. Originality/value In this paper and for the first time: the authors presented results on the numerical simulation for classes of time-fractional PDEs such as those found in the transonic multiphase flows. The authors applied the reproducing kernel method systematically for the numerical solutions of time-fractional Tricomi and Keldysh equations subject to Robin functions types.

A New Method for Computing Axisymmetric Buckling of Spherical Caps

1971 ◽  
Vol 38 (1) ◽  
pp. 179-184 ◽  
Author(s):  
G. A. Thurston
Keyword(s):  
Quadratic Form ◽  
Differential Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Nonlinear Differential Equations ◽  
Uniform Pressure ◽  
Axisymmetric Buckling ◽  
Spherical Caps ◽  
Limit Points ◽  
Check Experiment

A modification of Newton’s method is applied to the solution of the nonlinear differential equations for clamped, shallow spherical caps under uniform pressure. The linear form of Newton’s method or quasi-linearization breaks down at limit points of the differential equations. A simplified “quadratic form” is derived in the paper and shown to be satisfactory for continuing the solution past the limit point and into the postbuckling region. Results for the buckling pressures defined by the limit points agree with published results for perfect caps. New results are presented for imperfect caps that check experiment.

scholarly journals A New Method for Riccati Differential Equations Based on Reproducing Kernel and Quasilinearization Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
F. Z. Geng ◽  
X. M. Li
Keyword(s):  
Differential Equations ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Initial Position ◽  
New Method ◽  
Riccati Differential Equation ◽  
Reproducing Kernel Method ◽  
Riccati Differential Equations ◽  
Linear Problems ◽  
Quasilinearization Technique

We introduce a new method for solving Riccati differential equations, which is based on reproducing kernel method and quasilinearization technique. The quasilinearization technique is used to reduce the Riccati differential equation to a sequence of linear problems. The resulting sets of differential equations are treated by using reproducing kernel method. The solutions of Riccati differential equations obtained using many existing methods give good approximations only in the neighborhood of the initial position. However, the solutions obtained using the present method give good approximations in a larger interval, rather than a local vicinity of the initial position. Numerical results compared with other methods show that the method is simple and effective.

scholarly journals A NEW NUMERICAL METHOD TO SOLVE NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 26 (3) ◽  
pp. 469-478
Author(s):  
Jinjiao Hou ◽  
Jing Niu ◽  
Welreach Ngolo
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Nonlinear Problems ◽  
Reproducing Kernel Method ◽  
Homotopy Perturbation ◽  
Linear Problems

In this paper, a new method combining the simplified reproducing kernel method (SRKM) and the homotopy perturbation method (HPM) to solve the nonlinear Volterra-Fredholm integro-differential equations (V-FIDE) is proposed. Firstly the HPM can convert nonlinear problems into linear problems. After that we use the SRKM to solve the linear problems. Secondly, we prove the uniform convergence of the approximate solution. Finally, some numerical calculations are proposed to verify the effectiveness of the approach.

scholarly journals Solving Delay Differential Equations by an Accurate Method with Interpolation

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Ali Akgül ◽  
Adem Kiliçman
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Approximation ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Approximate Solutions ◽  
Accurate Method ◽  
Reproducing Kernel Method ◽  
Comparison Of The Results ◽  
Delay Differential

We use the reproducing kernel method (RKM) with interpolation for finding approximate solutions of delay differential equations. Interpolation for delay differential equations has not been used by this method till now. The numerical approximation to the exact solution is computed. The comparison of the results with exact ones is made to confirm the validity and efficiency.

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