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Iterative methods for nonlinear systems associated with finite difference approach in stochastic differential equations

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Abstract

We present some iterative methods of different convergence orders for solving systems of nonlinear equations. Their computational complexities are studies. Then, we introduce the method of finite difference for solving stochastic differential equations of Itô-type. Subsequently, our multi-step iterative schemes are employed in this procedure. Several experiments are finally taken into account to show that the presented approach and methods work well.

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References

  1. Atalla, M.A.: Finite-difference approximations for stochastic differential equations, Probabilistic methods for the investigation of systems with an infinite number of degrees of freedom. Inst. Math. Acad. Science USSR, Kiev, 11–16 (1986)

  2. Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: A modified Newton-Jarratt’s composition. Numer. Algor. 55, 87–99 (2010)

    Article  MATH  Google Scholar 

  3. Fornberg, B.: Calculation of weights in finite difference formulas. SIAM Rev. 40, 685–691 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fornberg, B.: Finite difference method. Scholarpedia 6(10), 9685 (2011)

    Article  Google Scholar 

  5. Grau-Sánchez, M., Grau, À., Noguera, M.: On the computational efficiency index and some iterative methods for solving systems of nonlinear equations. J. Comput. Appl. Math. 236, 1259–1266 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Grau-Sánchez, M., Noguera, M., Amat, S.: On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods. J. Comput. Appl. Math. 237, 363–372 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Homeier, H.H.H.: On Newton-type methods with cubic convergence. J. Comput. Appl. Math. 176, 425–432 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Iacus, S.M.: Simulation and Inference for Stochastic Differential Equations with R Examples. Springer, New York, USA (2008)

    Book  MATH  Google Scholar 

  9. Kloeden, P.E., Platen, E., Schurz, H.: Numerical Solution of SDE through Computer Experiments. Springer-Verlag, Germany (2003)

    Google Scholar 

  10. Lamperti, J.: Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104, 62–78 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  11. Montazeri, H., Soleymani, F., Shateyi, S., Motsa, S.S.: On a new method for computing the numerical solution of systems of nonlinear equations. J. Appl. Math. 2012, 15. Article ID 751975

  12. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  13. Platen, E., Bruti-Liberati, N.: Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer-Verlag, Berlin, Heidelberg (2010)

    Book  MATH  Google Scholar 

  14. Sharma, J.R., Guha, R.K., Sharma, R.: An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer. Algor. 62, 307–323 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)

    MATH  Google Scholar 

  16. Ullah, M.Z., Soleymani, F., Al-Fhaid, A.S.: Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs. Numer. Algor. 67, 223–242 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wolfram Mathematica, Tutorial Collection : Advanced Numerical Differential Equation Solving in Mathematica. ISBN: 978-1-57955-058-5 (2008)

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Authors and Affiliations

  1. Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

    Ali R. Soheili & Fazlollah Soleymani

  2. The Center of Excellence on Modelling and Control Systems, Ferdowsi University of Mashhad, Mashhad, Iran

    Ali R. Soheili

Authors
  1. Ali R. Soheili
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  2. Fazlollah Soleymani
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Correspondence to Fazlollah Soleymani.

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Soheili, A.R., Soleymani, F. Iterative methods for nonlinear systems associated with finite difference approach in stochastic differential equations. Numer Algor 71, 89–102 (2016). https://doi.org/10.1007/s11075-015-9986-5

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  • DOI: https://doi.org/10.1007/s11075-015-9986-5

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