Abstract
The current paper contributes a new numerical algorithm for solving a class of second-order matrix differential equations. To do so, the operational matrix of integration based on the shifted Legendre polynomials together with the collocation method is used to reduce the main problem to coupled matrix equations. An error estimation is provided which verifies the exponential rate of convergence. Numerical experiments are reported to demonstrate the applicability and efficiency of the suggested scheme.
Similar content being viewed by others
References
Bernstein DS (2018) Scalar, vector and matrix mathematics. Theory, facts and formulas. Princeton University Press, New Jersey
Bhrawy AH, Abdelkawy MA, Ezz-Eldien SS (2016a) Efficient spectral collocation algorithm for a two-sided space fractional Boussinesq equation with non-local conditions. Mediterr J Math 13(5):2483–2506
Bhrawy AH, Doha EH, Ezz-Eldien SS, Abdelkawy MA (2016b) A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equations. Calcolo 53(1):1–17
Borhanifar A, Abazari R (2007) Numerical solution of second-order matrix differential models using cubic matrix splines. Appl Math Sci 1(59):2927–2937
Canuto C, Hussaini MY, Quarteroni A, Zang TA (1988) Spectral Methods in Fluid Dynamics. Springer, New York
Chang RY, Wang ML (1983) Shifted Legendre direct method for variational problems. J. Optim. Theory Appl. 39(2):299–307
Chen Y, Tang T (2010) Convergence analysis of the Jacobi spectral collocation methods for Volterra integral equations with a weakly singular kernel. Math. Comput. 79(269):147–167
Defez E, Soler L, Hervás A, Santamaría C (2005) Numerical solution of matrix differential models using cubic matrix splines. Comput Math Appl 50(5–6):693–699
Defez E, Hervás A, Soler L, Tung MM (2007) Numerical solutions of matrix differential models cubic spline II. Math Comput Model 46(5–6):657–669
Defez E, Hervás A, Ibáñez J, Tung MM (2012a) Numerical solutions of matrix differential models using higher-order matrix splines. Mediterr J Math 9(4):865–882
Defez E, Tung MM, Ibáñez JJ, Sastre J (2012b) Approximating and computing nonlinear matrix differential models. Math Comput Model 55(7–8):2012–2022
Dubey R, Vandana, Mishra VN (2018) Second order multiobjective symmetric programming problem and duality relations under \((F,G_{f})\)-convexity. Glob J Eng Sci Res 5(8):187–199
Ezz-Eldien SS (2016) New quadrature approach based on operational matrix for solving a class of fractional variational problems. J Comput Phys 317:362–381
Ezz-Eldien SS (2018a) On solving fractional logistic population models with applications. Comput Appl Math. 37(5):6392–6409. https://doi.org/10.1007/s40314-018-0693-4
Ezz-Eldien SS (2018b) On solving systems of multi-pantograph equations via spectral tau method. Appl Math Comput 321:63–73
Ezz-Eldien SS, Doha EH (2018) Fast and precise spectral method for solving pantograph type Volterra integro-differential equations. Numer Algorithm. https://doi.org/10.1007/s11075-018-0535-x
Ezz-Eldien SS, El-Kalaawy AA (2018) Numerical simulation and convergence analysis of fractional optimization problems with right-sided Caputo fractional derivative. J Comput Nonlinear Dyn 13(1):011010
Ezz-Eldien SS, Hafez RM, Bhrawy AH, Baleanu D, El-Kalaawy AA (2017) New numerical approach for fractional variational problems using shifted Legendre orthonormal polynomials. J Optim Theory Appl 174(1):295–320
Ezz-Eldien SS, Doha EH, Bhrawy AH, El-Kalaawy AA, Machadod JAT (2018) A new operational approach for solving fractional variational problems depending on indefinite integrals. Commun Nonlinear Sci Numer Simul 57:246–263
Flett TM (1980) Differential analysis: differentiation, differential equations and differential inequalities. Cambridge University Press, Cambridge
Hafez RM, Ezz-Eldien SS, Bhrawy AH, Ahmed EA, Baleanu D (2015) A Jacobi Gauss–Lobatto and Gauss–Radau collocation algorithm for solving fractional Fokker–Planck equations. Nonlinear Dyn 82(3):1431–1440
Maleknejad K, Nouri K, Torkzadeh L (2016) Operational matrix of fractional integration based on the shifted second kind Chebyshev polynomials for solving fractional differential equations. Mediterr J Math 13(3):1377–1390
Maleknejad K, Nouri K, Torkzadeh L (2017) Study on multi-order fractional differential equations via operational matrix of hybrid basis functions. Bull Iran Math Soc 43(2):307–318
Mishra LN (2017) Scalar, on existence and behavior of solutions to some nonlinear integral equations with Applications. Ph.D. Thesis, National Institute of Technology, Silchar 788 010, Assam, India (2017)
Mishra VN, Mishra LN (2012) Trigonometric approximation of signals (functions) in \(L_p\)-norm. Int J Contemp Math Sci 7(19):909–918
Negarchi N, Nouri K (2018) Numerical solution of Volterra–Fredholm integral equations using the collocation method based on a special form of the Müntz–Legendre polynomials. J Comput Appl Math 344:15–24
Pantelousa AA, Karageorgosc AD, Kalogeropoulosc GI (2014) A new approach for second-order linear matrix descriptor differential equations of Apostol–Kolodner type. Math Methods Appl Sci 37(2):257–264
Pishbin S, Ghoreishi F, Hadizadeh M (2011) A posteriori error estimation for the Legendre collocation method applied to integral-algebraic Volterra equations. Electron Trans Numer Anal 38:327–346
Rivlin TJ (1969) An introduction to the approximation of functions. Blaisdell Publishing Company, Waltham
Tang T, Xu X, Cheng J (2008) On spectral methods for Volterra type integral equations and the convergence analysis. J Comput Math 26(6):825–837
Tung MM, Defez E, Sastre J (2008) Numerical solutions of second-order matrix models using cubic-matrix splines. Comput Math Appl 56(10):2561–2571
Vandana (2017) A study of dynamic inventory involving economic ordering of commodity. Ph.D. Thesis, Pt. Ravishankar Shukla University Raipur, 492010, Chhattisgarh, India
Vandana, Dubey R, Deepmala, Mishra LN, Mishra VN (2018) Duality relations for a class of a multiobjective fractional programming problem involving support functions. Am J Oper Res 8:294–311
Zhao J, Xiao J, Ford NJ (2014) Collocation methods for fractional integro-differential equations with weakly singular kernels. Numer Algorithm 65(4):723–743
Zhou B, Cai GB, Duan GR (2013) Stabilisation of time-varying linear systems via Lyapunov differential equations. Int J Control 86(2):332–347
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nouri, K., Panjeh Ali Beik, S. & Torkzadeh, L. Operational Matrix Approach for Second-Order Matrix Differential Models. Iran J Sci Technol Trans Sci 43, 1925–1932 (2019). https://doi.org/10.1007/s40995-018-0666-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40995-018-0666-x
Keywords
- Matrix differential equation
- Shifted Legendre polynomials
- Operational matrix of integration
- Collocation method
- Error estimation