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 Reproducing Kernel Method for Fractional Riccati Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
X. Y. Li ◽  
B. Y. Wu ◽  
R. T. Wang
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Present Method ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Initial Position ◽  
Numerical Results ◽  
Reproducing Kernel Method ◽  
Riccati Differential Equations ◽  
Quasilinearization Technique

This paper is devoted to a new numerical method for fractional Riccati differential equations. The method combines the reproducing kernel method and the quasilinearization technique. Its main advantage is that it can produce good approximations in a larger interval, rather than a local vicinity of the initial position. Numerical results are compared with some existing methods to show the accuracy and effectiveness of the present method.

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.

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.

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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