An operational matrix method is a well known method to solve an initial value problem (IVP). A recent contribution to the operational matrix method is frame operational matrix (FOM) method. The shooting method is a standard method which converts a boundary value problem (BVP) into an IVP. The BVP is then solved by solving the IVP. In this paper, we combine the shooting method with that of the FOM method to solve two point BVPs. In short, this method is described as the following. First, we convert the BVP into an IVP by shooting method. The IVP is then solved by the FOM method. The approximate solution thus obtained is checked whether it satisfy both the boundary condition. If not, then we take better approximations until they satisfy both the boundary condition. To get a better approximation, we use the Newton’s iteration method.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Ahsan, M., Farrukh, S.: A new type of shooting method for nonlinear boundary value problems. Alex. Eng. J. 52, 801–805 (2013)
Alijani, Z., Baleanu, D., Shiri, B., Wu, G.C.: Spline collocation methods for systems of fuzzy fractional differential equations. Chaos Soliton Fractals 131, 109510 (2020)
Atkinson, K.E.: An Introductiobn To Numerical Analysis. Wiley, Hoboken (1989)
Aydogan, S.M., Baleanu, D., Mohammadi, H., Rezapour, S.: On the mathematical model of Rabies by using the fractional Caputo–Fabrizio derivative. Adv. Differ. Equ. 382, 1–21 (2020)
Baleanu, D., Shiri, B.: Collocation methods for fractional differential equations involving non-singular kernel. Chaos Soliton Fractals 116, 136–145 (2018)
Baleanu, D., Etemad, S., Rezapour, S.: A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions. Bound. Value Probl. 64, 1–16 (2020)
Baleanu, D., Mohammadi, H., Rezapour, S.: A fractional differential equation model for the COVID-19 transmission by using the Caputo–Fabrizio derivative. Adv. Differ. Equ. 299, 1–27 (2020)
Baleanu, D., Mohammadi, H., Rezapour, S.: A mathematical theoretical study of a particular system of Caputo-Fabrizio fractional differential equations for the Rubella disease model. Adv. Differ. Equ. 184, 1–19 (2020)
Baleanu, D., Mohammadi, H., Rezapour, S.: Analysis of the model of HIV-1 infection of CD4+ T-cell with a new approach of fractional derivative. Adv Differ Equ 71, 1–17 (2020)
Bulirsch, R.: Die Mehrzielmethode zur Numerischen Losung von Nichtlinearen Randwertproblemen und Aufgaben der Optimalen Steuerung, Vortrag im Lehrgang “Flugbahnoptimierung” der Carl-Cranz-Gesellschaft, West Germany (1971)
Bulirsch, R., Stoer, J., Deuflhard, P.: Numerical Solution of Nonlinear Two-Point Boundary Value Problems. Numerische Mathematik, L Handbook Series Approximation (in press) (1978)
Chen, C., Hsiao, C.H.: Haar wavelet method for solving lumped and distributed parameter system. IEE Proc. Control Theory Appl. 144, 87–94 (1997)
Christensen, O.: An Introduction to Frames and Riesz Basis. Birkhauser, Berlin (2003)
Chui, C.K.: An Introduction to Wavelets. Academic Press, San Diego (1992)
Chui, C.K., Wenjie, H.: Compactly supported tight frames associated with refinable functions. Appl. Comput. Harmon. Anal. 8, 293–319 (2000)
Dadkhah, E., Shiri, B., Ghaffarzadeh, H., Baleanu, D.: Visco-elastic dampers in structural buildings and numerical solution with spline collocation methods. J. Appl. Math. Comput. 63, 29–57 (2019)
Daubechies, I.: Ten Lectures on Wavelet. SIAM, Philadelphia (1992)
Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier Series. Trans. Am. Math. Soc. 72, 341–366 (1952)
EI-Kalla, I.L.: Piecewise continuous solution to a class of nonlinear boundary value problem, Ain Shams Eng. J., 4, 325–331(2013)
Goodman, T.R., Lance, G.N.: The numerical solution of two-point boundary value problems. Math. Tables Other Aids Comput. 10, 82–86 (1956)
Jena, M.K., Sahu, K.S.: Haar wavelet operational matrix method to solve initial value problems: a short survey. Int. J. Appl. Comput. Math. 3, 3961–3975 (2017)
Jena, M.K., Sahu, K.S.: Operational matrices from a frame and their applications in solving boundary value problems with mixed boundary conditions. Int. J. Appl. Comput. Math. 4, 127 (2018)
Keller, H.B.: Numerical methods for two point boundary value problems. Ginn-Blaisdell, Waltham (1968)
Keller, H.B.: Numerical Solution of Two Point Boundary Value Problems. SIAM, Philadelphia (1976)
Khiabani, E.D., Ghaffarzadeh, H., Shiri, B., Katebi, J.: Spline collocation methods for seismic analysis of multiple degree of freedom systems with visco-elastic dampers using fractional models. J. Vib. Control 26, 1445–1465 (2020)
Lepik, U.: Numerical solution of differential equations using Haar wavelets. Math. Comput. Simul. 68, 127–143 (2005)
Lepik, U.: Solving fractional integral equations by the Haar wavelet method. Appl. Math. Comput. 214, 468–478 (2009)
Lepik, U., Hein, H.: Haar Wavelets with Applications. Springer, Berlin (2014)
Liu, C.: Efficient shooting methods for second order ordinary differential equations. Comput. Model. Eng. Sci. 15, 69–86 (2006)
Matinfar, M., Ghasemi, M.: Solving BVPs with shooting method and VIMHP. J. Egypt. Math. Soc. 21, 354–360 (2013)
Matthij, R.M., Staarink, G.: An efficient algorithm for solving general linear two-point BVP. SIAM J. Sci. Comput. 5, 745–763 (1984)
Mehra, M.: Wavelet-Galerkin Methods. In: Wavelets Theory and its Applications. Forum for Interdisciplinary Mathematics. Springer, Singapore, pp. 121–133 (2018)
Morrison, D.D., Riley, J.D., Zancanaro, J.F.: Multiple shooting method for two-point boundary value problems. Commun. ACM 5, 613–614 (1962)
Rezapour, S., Mohammadi, H., Jajarmi, A.: A new mathematical model for Zika virus transmission. Adv Differ. Equ. 589, 1–5 (2020)
Roberts, S.M., Shipman, J.S.: Continuation in shooting methods for two-point boundary value problems. J. Math. Anal. Appl. 18, 45–58 (1967)
Sahu, K.S., Jena, M.K.: Solution of initial value problems using an operational matrix. Int. J. Appl. Comput. Math. 6(61), 1–23 (2020)
Shiri, B., Baleanu, D.: System of fractional differential algebraic equations with applications. Chaos Soliton Fractals 120, 203–212 (2019)
Tuan, N.H., Mohammadi, H., Rezapour, S.: A mathematical model for COVID-19 transmission by using the Caputo fractional derivative. Chaos Soliton Fractals 140, 1–11 (2020)
Xu, J.C., Shann, W.C.: Galerkin-wavelet methods for two-point boundary value problems. Numer. Math. 63, 123–144 (1992)
You Ma, C., Shiri, B., Wu, G.C., Baleanu, D.: New fractional signal smoothing equations with short memory and variable order. Optik 218, 164507 (2020)
We would like to express our great appreciation to the reviewers for their valuable and constructive suggestions which improved the quality of the article.
Conflict of interest
The authors declare that they have no conflict of interest.
KSS and MKJ formulated this research to solve boundary value problem. The major part of this research is carried out by KSS.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Below is the link to the electronic supplementary material.
Frame Matrix for \(J=0\left( M=3\right) \)
Operational Matrix for \(J=0\left( M=3\right) \)
Frame Matrix for \(J=1\left( M=7\right) \)
Operational Matrix for \(J=1\left( M=7\right) \)
About this article
Cite this article
Sahu, K.S., Jena, M.K. Combining the Shooting Method with an Operational Matrix Method to Solve Two Point Boundary Value Problems. Int. J. Appl. Comput. Math 7, 29 (2021). https://doi.org/10.1007/s40819-021-00967-x
- Boundary value problems
- Initial value problems
- Operational matrix
- Shooting method
Mathematics Subject Classification