# THIRD ORDER ITERATIVE METHOD FOR SOLVING NON-LINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATION IN FINANCIAL APPLICATION

## Authors

• Kedir Aliyi Koroche Department of Mathematics, College of Natural and Computational Sciences, Ambo University, Ambo, Ethiopia https://orcid.org/0000-0003-3769-4870

## Keywords:

Non-Linear Parabolic Equation, Third Order Iterative Scheme, Convergent Analysis

## Abstract

In this paper, third order iterative scheme is presented for working the solution the non-linear stochastic parabolic equation in one dimensional space. First, the given result sphere is discretized by using invariant discretization grid point. Next, by using Taylor series expansion we gain the discretization of the model problem. From this, we gain the system of nonlinear ordinary difference equations.  By rearranging this scheme, we gain iterative schemes which is called gauss Jacobean iterative scheme. To validate the convergences of the proposed system, three model illustrations are considered and answered it at each specific grid point on its result sphere. The coincident (convergent) analysis of the present techniques is worked by supported the theoretical and fine statements and the delicacy of the result is attained. The delicacy of the present techniques has been shown in the sense of average absolute error (AAE), root mean square error norm and point-wise maximum absolute error norm and comparing gets crimes in the result attained in literature and these results are also presented in tables and graphs. The physical gets of results between numerical versus are also been presented in terms of graphs. As we can see from the table and graphs, the present system approach are approximates the exact result veritably well and it's relatively effective and virtually well suited for working the solution for non-linear parabolic equation.

## References

Ahmed H., (2017) Faedo-Galerkin method for heat equation, Global J. of Pure and App. Math. ,13(4), 1195-1207.

Alharbi R. (2020), Nonlinear Parabolic Stochastic Partial Differential Equation with Application to Finance, A University of Sussex PhD thesis.

Aliyi K., Shiferaw A., and Muleta H. (2021). Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation, American J. of Math. And Comp. Mod., 6(2), pp.35-42. Retrieved from https://doi.org/10.11648/j.ajmcm.20210602.12

Ambrosio L., Caffarelli L. A., Brenier Y., Buttazzo G., and Villani C. (2001). Optimal Transportation and Applications : Lectures from the C.I.M.E. Summer School held in Martina Franca.

Behzadi S. S. and Araghi M. A. F. (2011). Numerical solution for solving Burger's-Fisher equation by using iterative methods, Math. Comput. Appl. 16(2), 443-455. Retrieved from https://doi.org/10.3390/mca16020443

Evans L. C (1998). Partial Differential Equations, Grad. Stud. Math. AMS. Providence. RI.

Feng X, Glowinski R., and Neilan M. (2013), Recent developments in numerical methods for fully nonlinear second-order partial differential equations, Siam. rev. vol. 52, pp. 205-67. Retrieved from https://doi.org/10.1137/110825960

Gatheral J., and Taleb N. N. (2013). The Volatility surface A Practitioner's Guide. John Wiley & Sons, New York, NY. (2011). OCLC : 899182374. Retrieved from https://doi.org/10.1002/9781119202073

Hepson O. E. (2021). An exponential cubic B-spline algorithm for solving the nonlinear Coupled Burgers' equation, Comp. Methods for Dif. Eq.,9, (4), pp.1109-1127.

JAIN M. K., IYENGAR S. R. K., and JAIN R. K. (1984). Numerical Methods for Scientific and engineering Computation, Published in the Western Hemisphere by Halsted Press, A Division of John Wiley & Sons, Inc., New York, New Delhi October

Kocacoban D., Koc A. B., Kurnaz A., and Keskin Y., (2011). A better approximation to the solution of Burger-Fisher equation, in Proceedings of the World Congress on Engineering 1, 1-6.

Li Y., Shu C. W., and Tang S. (2021). A Local Discontinuous Galerkin Method For Nonlinear Parabolic SPDES, ESAIM Math. Mod., and Num. Anlys. Vol.55, pp. S187-S223. Retrieved from https://doi.org/10.1051/m2an/2020026

Lima A. S., Kamrujjaman M., and Islam M. S. (2021). Numerical solution of convection-diffusion-reaction Equations by a finite element method with error correlation, AIP Advances, vol.11, 085225. Retrieved from https://doi.org/10.1063/5.0050792

Maher A., El-Hawary H. M., and Al-Amry M. S. (2013). New Exact Solutions for New Model Nonlinear Partial Differential Equation, J. of Appl. Math., 2013. Retrieved from https://doi.org/10.1155/2013/767380

Mohanty R. K., and Jha N. (2005). A class of variable meshes spline in compression methods for singularly perturbed two-point singular boundary-value problems, Appl. Math. &Comp., Vol.168, 704-716. Retrieved from https://doi.org/10.1016/j.amc.2004.09.049

Morton K.W., and Mayers D. F (2005). Numerical Solution of Partial Differential Equations, An introduction, Second Edition, Cambridge University Press, New York. Retrieved from https://doi.org/10.1017/CBO9780511812248

Ngoc L. T. P., Son L. H. K., and Long N. T. (2017). An N-order iterative scheme for a nonlinear Carrier wave in an annular with Robin-Dirichlet conditions, Nonlinear Functional Analysis and applications, 22 (1), pp.147-169, Retrieved from https://digital.lib.ueh.edu.vn/handle/UEH/56185

Ngoc L. T. P., Truong L. X., and Long N. T. (2010). High-order iterative methods for a nonlinear Kirchhoff wave align, Demonstrations Mathematica, 43(3) ,pp. 605-634. Retrieved from https://doi.org/10.1515/dema-2010-0310

Nhan N. H., Dung T. T. M, Thanh L. T. M., Ngoc L. T. P., and Long N. T. (2021). A High-Order Iterative Scheme for a Nonlinear Pseudo-parabolic Equation and Numerical Results. Mathematical Problems in Engineering. Retrieved from https://doi.org/10.1155/2021/8886184

Orlando G., Mininni R. M., and Bufalo M (2018). A New Approach to CIR Short-Term Rates Modelling. New Methods in Fixed Income Modeling, Contributions to management Science. Springer International Publishing, pp. 35- 43. Retrieved from https://doi.org/10.1007/978-3-319-95285-7_2

Orlando G., Mininni R.M., and Bufalo M. (1 Jan. 2019), A new approach to forecast market interest rate through the CIR model, Studies in Economics and Finance, 267-292. Retrieved from https://doi.org/10.1108/SEF-03-2019-0116

Pardoux E. (2007), Stochastic Partial Differential Equations. Lecture notes for the course given at Fudan University, Shanghai.

Prevot C., and Rockner M. (2007). A Concise Course on Stochastic Partial Differential Equations. In : Vol. 1905 of Lecture Notes in Mathematics. Springer, Berlin.

Rouah F., (n.d.) Hesston Model and its Extensions in Matlab and C. Hoboken, Wiley.

Taylor M. E. (1996), Partial Differential Equations II : Qualitative Studies of Linear Equations, Appl. Math. Sci. Springer, New York.

Taylor M. E. (1996). Partial Differential Equations I : Basic Theory, Appl. Math. Sci. Springer, New York.

Truong L. X., Ngoc L. T. P., and Long N. T. (2009). The n-order iterative schemes for a nonlinear Kirchhoff-Carrier wave equation associated with the mixed inhomogeneous conditions, Applied Mathematics and Computation, 215(5), pp. 1908-1925. Retrieved from https://doi.org/10.1016/j.amc.2009.07.056

Truong L. X., Ngoc L. T. P., and Long N. T. (2009). High-order iterative schemes for a nonlinear Kirchhoff-Carrier wave align associated with the mixed homogeneous conditions, Nonlinear Analysis : theory, Methods & Applications, 71(1), pp. 467-484. Retrieved from https://doi.org/10.1016/j.na.2008.10.086

Yavuz M. (2018). Novel solution methods for initial boundary value problems of fractional order with conformable differentiation, An International J. of Opt. and Cont. Theory. And appl.,8(1), pp.1-7. Retrieved from https://doi.org/10.11121/ijocta.01.2018.00540

Yavuz M. and. Ozdemir N. (2017). New numerical techniques for solving fractional partial differential equations in conformable sense, in Non-integer Order Calculus and its Applications, pp. 49-62. Retrieved from

https://doi.org/10.1007/978-3-319-78458-8_5Ahmed H., (2017) Faedo-Galerkin method for heat equation, Global J. of Pure and App. Math. ,13(4), 1195-1207.

Alharbi R. (2020), Nonlinear Parabolic Stochastic Partial Differential Equation with Application to Finance, A University of Sussex PhD thesis.

Aliyi K., Shiferaw A., and Muleta H. (2021). Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation, American J. of Math. And Comp. Mod., 6(2), pp.35-42. Retrieved from https://doi.org/10.11648/j.ajmcm.20210602.12

Ambrosio L., Caffarelli L. A., Brenier Y., Buttazzo G., and Villani C. (2001). Optimal Transportation and Applications : Lectures from the C.I.M.E. Summer School held in Martina Franca.

Behzadi S. S. and Araghi M. A. F. (2011). Numerical solution for solving Burger's-Fisher equation by using iterative methods, Math. Comput. Appl. 16(2), 443-455. Retrieved from https://doi.org/10.3390/mca16020443

Evans L. C (1998). Partial Differential Equations, Grad. Stud. Math. AMS. Providence. RI.

Feng X, Glowinski R., and Neilan M. (2013), Recent developments in numerical methods for fully nonlinear second-order partial differential equations, Siam. rev. vol. 52, pp. 205-67. Retrieved from https://doi.org/10.1137/110825960

Gatheral J., and Taleb N. N. (2013). The Volatility surface A Practitioner's Guide. John Wiley & Sons, New York, NY. (2011). OCLC : 899182374. Retrieved from https://doi.org/10.1002/9781119202073

Hepson O. E. (2021). An exponential cubic B-spline algorithm for solving the nonlinear Coupled Burgers' equation, Comp. Methods for Dif. Eq.,9, (4), pp.1109-1127.

JAIN M. K., IYENGAR S. R. K., and JAIN R. K. (1984). Numerical Methods for Scientific and engineering Computation, Published in the Western Hemisphere by Halsted Press, A Division of John Wiley & Sons, Inc., New York, New Delhi October

Kocacoban D., Koc A. B., Kurnaz A., and Keskin Y., (2011). A better approximation to the solution of Burger-Fisher equation, in Proceedings of the World Congress on Engineering 1, 1-6.

Li Y., Shu C. W., and Tang S. (2021). A Local Discontinuous Galerkin Method For Nonlinear Parabolic SPDES, ESAIM Math. Mod., and Num. Anlys. Vol.55, pp. S187-S223. Retrieved from https://doi.org/10.1051/m2an/2020026

Lima A. S., Kamrujjaman M., and Islam M. S. (2021). Numerical solution of convection-diffusion-reaction Equations by a finite element method with error correlation, AIP Advances, vol.11, 085225. Retrieved from https://doi.org/10.1063/5.0050792

Maher A., El-Hawary H. M., and Al-Amry M. S. (2013). New Exact Solutions for New Model Nonlinear Partial Differential Equation, J. of Appl. Math., 2013. Retrieved from https://doi.org/10.1155/2013/767380

Mohanty R. K., and Jha N. (2005). A class of variable meshes spline in compression methods for singularly perturbed two-point singular boundary-value problems, Appl. Math. &Comp., Vol.168, 704-716. Retrieved from https://doi.org/10.1016/j.amc.2004.09.049

Morton K.W., and Mayers D. F (2005). Numerical Solution of Partial Differential Equations, An introduction, Second Edition, Cambridge University Press, New York. Retrieved from https://doi.org/10.1017/CBO9780511812248

Ngoc L. T. P., Son L. H. K., and Long N. T. (2017). An N-order iterative scheme for a nonlinear Carrier wave in an annular with Robin-Dirichlet conditions, Nonlinear Functional Analysis and applications, 22 (1), pp.147-169, Retrieved from https://digital.lib.ueh.edu.vn/handle/UEH/56185

Ngoc L. T. P., Truong L. X., and Long N. T. (2010). High-order iterative methods for a nonlinear Kirchhoff wave align, Demonstrations Mathematica, 43(3) ,pp. 605-634. Retrieved from https://doi.org/10.1515/dema-2010-0310

Nhan N. H., Dung T. T. M, Thanh L. T. M., Ngoc L. T. P., and Long N. T. (2021). A High-Order Iterative Scheme for a Nonlinear Pseudo-parabolic Equation and Numerical Results. Mathematical Problems in Engineering. Retrieved from https://doi.org/10.1155/2021/8886184

Orlando G., Mininni R. M., and Bufalo M (2018). A New Approach to CIR Short-Term Rates Modelling. New Methods in Fixed Income Modeling, Contributions to management Science. Springer International Publishing, pp. 35- 43. Retrieved from https://doi.org/10.1007/978-3-319-95285-7_2

Orlando G., Mininni R.M., and Bufalo M. (1 Jan. 2019), A new approach to forecast market interest rate through the CIR model, Studies in Economics and Finance, 267-292. Retrieved from https://doi.org/10.1108/SEF-03-2019-0116

Pardoux E. (2007), Stochastic Partial Differential Equations. Lecture notes for the course given at Fudan University, Shanghai.

Prevot C., and Rockner M. (2007). A Concise Course on Stochastic Partial Differential Equations. In : Vol. 1905 of Lecture Notes in Mathematics. Springer, Berlin.

Rouah F., (n.d.) Hesston Model and its Extensions in Matlab and C. Hoboken, Wiley.

Taylor M. E. (1996), Partial Differential Equations II : Qualitative Studies of Linear Equations, Appl. Math. Sci. Springer, New York.

Taylor M. E. (1996). Partial Differential Equations I : Basic Theory, Appl. Math. Sci. Springer, New York.

Truong L. X., Ngoc L. T. P., and Long N. T. (2009). The n-order iterative schemes for a nonlinear Kirchhoff-Carrier wave equation associated with the mixed inhomogeneous conditions, Applied Mathematics and Computation, 215(5), pp. 1908-1925. Retrieved from https://doi.org/10.1016/j.amc.2009.07.056

Truong L. X., Ngoc L. T. P., and Long N. T. (2009). High-order iterative schemes for a nonlinear Kirchhoff-Carrier wave align associated with the mixed homogeneous conditions, Nonlinear Analysis : theory, Methods & Applications, 71(1), pp. 467-484. Retrieved from https://doi.org/10.1016/j.na.2008.10.086

Yavuz M. (2018). Novel solution methods for initial boundary value problems of fractional order with conformable differentiation, An International J. of Opt. and Cont. Theory. And appl.,8(1), pp.1-7. Retrieved from https://doi.org/10.11121/ijocta.01.2018.00540

Yavuz M. and. Ozdemir N. (2017). New numerical techniques for solving fractional partial differential equations in conformable sense, in Non-integer Order Calculus and its Applications, pp. 49-62. Retrieved from

2022-03-27

## How to Cite

Koroche, K. A. (2022). THIRD ORDER ITERATIVE METHOD FOR SOLVING NON-LINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATION IN FINANCIAL APPLICATION. International Journal of Engineering Science Technologies, 6(2), 21–37. https://doi.org/10.29121/ijoest.v6.i2.2022.299

Articles