A non-local boundary value problem with Caputo fractional derivative of order \(1<\nu <2\) is considered in this article. A numerical method comprising of an upwind difference scheme which is used to approximate the convection term and an \(L_2\) approximation of Caputo fractional derivative on an uniform mesh is constructed. Error estimate is derived. Numerical results are presented which validate our numerical 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.
Oustaloup, A.: Systems Asservis Linaires d’ordre Fractionnaire. Masson, Paris (1983)
Magin, R.: Fractional Calculus in Bioengineering. Begell House Publishers, Redding (2006)
Kaur, A., Takhar, S., Smith, M., Mann, E., Brashears, M.M.: Fractional differential equations based modeling of microbial survival and growth curves: model development and experimental validation. Food Eng. Phys. Properties 73, 403–414 (2008)
Diethelm, Kai: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)
Cabada, A., Wanassi, O.K.: Existence results for nonlinear fractional problems with non homogeneous integral boundary conditions. Math. Comput. Sci. 8, 1–15 (2020)
Cabada, A., Aleksić, S., Tomović, T.V., Dimitrijević, S.: Existence of solutions of nonlinear and non-local fractional boundary value problems. Mediterr. J. Math. 16, 1–20 (2019)
Chen, P., Gao, Y.: Positive solutions for a class of nonlinear fractional differential equations with nonlocal boundary value conditions. Positivity 22, 761–772 (2018)
Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta. Appl. Math. 109, 973–1033 (2010)
Chergui, D., Oussaeif, T.E., Ahcene, M.: Existence and uniqueness of solutions for nonlinear fractional differential equations depending on lower-order derivative with non-separated type integral boundary conditions. AIMS Math. 4, 112–133 (2019)
Gadzova, LKh: Nonlocal boundary value problem for a linear ordinary differential equation with fractional discretely distributed differentiation operator. Math. Notes 106, 904–908 (2019)
Li, P., Xu, C.: Boundary value problems of fractional order differential equation with integral boundary conditions and not instantaneous impulses. J. Function Spaces 2015, 904–908 (2015)
Arqub, O.A.: Numerical solutions of systems of first-order, two-point BVPs based on the reproducing kernel algorithm. Calcolo 55, 1–28 (2018)
Arqub, O.A.: Application of residual power series method for the solution of time-fractional Schrödinger equations in one-dimensional space. Fundamenta Inform. 166, 87–110 (2019)
Naik, P.A., Zu, J., Owolabi, K.M.: Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control. Chaos, Solitons and Fractals 138, 1–24 (2020)
Naik, P.A., Owolabi, K.M., Yavuz, M., Zu, J.: Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells. Chaos, Solitons and Fractals 140, 1–13 (2020)
Pedas, A., Tamme, E.: Piewise polynomial collocation for linear boundary value problems of fractional differential equations. J. Comput. Appl. Math. 236, 3349–3359 (2012)
Stynes, M., Gracia, J.L.: A finite difference method for a two-point boundary value problem with a Caputo fractional derivative. IMA J. Numer. Anal. 35, 1–24 (2014)
Gracia, J.L., Stynes, M.: Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems. J. Comput. Appl. Math. 273, 103–115 (2015)
Cen, Z., Huang, J., Xu, A.: An efficient numerical method for a two-point boundary value problem with a Caputo fractional derivative. J. Comput. Appl. Math. 336, 1–7 (2018)
Kopteva, N., Stynes, M.: An efficient collocation method for a Caputo two-point boundary value problem. BIT Numer. Math. 55, 1105–1123 (2015)
El-Ajou, A., Arqub, O., Momani, S.: Solving fractional two-point boundary value problems using continuous analytic method. Ain Shams Eng. J. 4, 539–547 (2013)
Santra, S., Mohapatra, J.: Analysis of the L1 scheme for a time fractional parabolic-elliptic problem involving weak singularity. Math. Meth. Appl. Sci. 44, 1–13 (2020)
Jin, B., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36, 1–25 (2015)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Brunner, H., Pedas, A., Vainikko, G.: Piecewise polynomial collocation methods for linear volterra integro-differential equations with weakly singular kernels. SIAM J. Numer. Anal. 39, 957–982 (2001)
Al-Refai, M.: Basic results on nonlinear eigenvalue problems of fractional order. Electron. J. Differ. Equ. 191, 1–12 (2012)
Sousa, E.: How to approximate the fractional derivative of order. Int. J. Bifurc. Chaos 22, 1–13 (2012)
Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Applied Mathematics. Chapman & Hall/CRC, Florida (2000)
The first authour wishes to thank Bharathidasan University for its financial support under URF scheme. The authors wish to thank Department of Science and Technology, Government of India, for the computing facility under DST- PURSE phase II Scheme.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Mary, S.J.C., Tamilselvan, A. Numerical method for a non-local boundary value problem with Caputo fractional order. J. Appl. Math. Comput. (2021). https://doi.org/10.1007/s12190-021-01501-4
- Fractional differential equation
- Caputo fractional derivative
- Non-local boundary value problem
- Maximum principle
- Finite difference scheme
- Error estimate
Mathematics Subject Classification