### SOLUTION OF SINGULAR RICCATI DIFFERENTIAL EQUATIONS USING THE REPRODUCING KERNEL HILBERT SPACE METHOD

#### Abstract

This paper deals with the approximation the solution of singular Riccati differential equations using the reproducing kernel Hilbert space scheme. The exact solution *u*(*r*) is represented in the form of series in the space . In the mean time, the *n*-term approximate solution *u*(*r*) obtained and is proved to converge to the exact solution *u*(*r*). Some numerical examples have also been studied to demonstrate the accuracy of the present method. Numerical experiments are performed to confirm our theoretic findings.

**2010 Mathematics Subject Classification:** 34*K*28; 34*K*07; 34*B*10

#### Keywords

#### Full Text:

PDF#### References

K. Maleknejad, A. Arzhang, Numerical solution of the Fredholm singular integro-differential equation with Cauchy kernel by using Taylor-series expansion and Galerkin method, Applied Mathematics and Computation 182 (2006) 888-897.

H. Brunner, A. Pedas, G. Vainikko, A spline collocation method for linear Volterra integro-differential equations with weakly singular kernels, BIT 41(5) (2001) 891-900.

P. M. Lima, A. Bellour, M. V. Bulatov, Numerical solution of integrodifferential equations arising from singular boundary value problems, Applied Mathematics and Computation 336 (2018) 1-15.

N. Aronszajn, Theory of reproducing kernels, Transactions of the American Mathematical Society, 68 (1950) 337-404.

M.G. Cui, Y.Z. Lin, Nonlinear Numerical Analysis in Reproducing Kernel Space, Nova Science Publisher, New York, 2009.

D. Alpay, Reproducing Kernel Spaces and Applications, Birkh¨auser, Berlin, 2003.

A. Berlinet, C. Thomas-Agnan, Reproducing Kernel Hilbert Space in Probability and Statistics, Kluwer Academic Publishers, 2004.

H. Beyrami, T. Lotfi, K. Mahdiani, A new efficient method with error analysis for solving the second kind Fredholm integral equation with Cauchy kernel, J. Comput. Appl. Math. 300 (2016) 385–399.

H. Beyrami, T. Lotfi, K. Mahdiani, Stability and error analysis of the reproducing kernel Hilbert space method for the solution of weakly singular Volterra integral equation on graded mesh, Applied Numerical Mathematics 120 (2017) 197-214.

Y. Wang, T. Chaolu, Z. Chen, Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems, Int. J. Comput. Math. 87 (2010) 367-380.

F.Z. Geng, M.G. Cui, Solving singular nonlinear two-point boundary value problems in the reproducing kernel space, J. Korean Math. Soc. 45(3) (2008) 77-87.

M.G. Cui and F.Z. Geng, Solving singular two-point boundary value problem in reproducing kernel space, J. Comput. Appl. Math. 205 (2007) 6-15.

M. Khaleghi, M.T. Moghaddam, E. Babolian, S. Abbasbandy, Solving a class of singular two-point boundary value problems using new effective reproducing kernel technique, Appl. Math. Comput. 331 (2018) 264-273.

F. Geng, A novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method, Appl. Math. Comput. 218 (2011) 4211-4215.

Y. Wang, L. Su, X. Cao, X. Li, Using reproducing kernel for solving a class of singularly perturbed problems, Comput. Math. Appl. 61 (2011) 421-430.

F.Z. Geng, S.P. Qian, S. Li, A numerical method for singularly perturbed turning point problems with an interior layer, J. Comput. Appl. Math. 225 (2014) 97-105.

F.Z. Geng, S.P. Qian, Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers, Appl. Math. Lett. 26 (2013) 998-1004.

F.Z. Geng, S.P. Qian, M.G. Cui, Improved reproducing kernel method for singularly perturbed differential-difference equations with boundary layer behavior, Appl. Math. Comput. 252 (2015) 58-63.

H. Sahihi, S, Abbasbandy, T. Allahviranloo, Reproducing kernel method for solving singularly perturbed differential-difference equations with boundary layer behavior in Hilbert space, J. Comput. Appl. Math. 328 (2018) 30-43.

H. Sahihi, S, Abbasbandy, T. Allahviranloo, Computational method based on reproducing kernel for solving singularly perturbed differential-difference equations with a delay, Appl. Math. Comput. 361 (2019) 583-598.

F. Geng, Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method, Appl. Math. Comput. 215 (2009) 2095-2102.

M.G. Sakar, Iterative reproducing kernel hilbert spaces method for Riccati differential equations, J. Comput. Appl. Math. 309 (2017) 163-174.

### Refbacks

- There are currently no refbacks.

This work is licensed under a Creative Commons Attribution 4.0 International License.

*Copyright © 2020*

**INTERNATIONAL EDUCATION AND RESEARCH JOURNAL**