Advertisement

Operational Matrix Approach for Second-Order Matrix Differential Models

  • Kazem Nouri
  • Samaneh Panjeh Ali Beik
  • Leila Torkzadeh
Research paper
  • 10 Downloads
Part of the following topical collections:
  1. Mathematics

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.

Keywords

Matrix differential equation Shifted Legendre polynomials Operational matrix of integration Collocation method Error estimation 

Mathematics Subject Classification

15A24 33D52 65F30 65L60 

References

  1. Bernstein DS (2018) Scalar, vector and matrix mathematics. Theory, facts and formulas. Princeton University Press, New JerseyCrossRefGoogle Scholar
  2. 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):2483–2506MathSciNetCrossRefGoogle Scholar
  3. 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):1–17MathSciNetCrossRefGoogle Scholar
  4. Borhanifar A, Abazari R (2007) Numerical solution of second-order matrix differential models using cubic matrix splines. Appl Math Sci 1(59):2927–2937MathSciNetzbMATHGoogle Scholar
  5. Canuto C, Hussaini MY, Quarteroni A, Zang TA (1988) Spectral Methods in Fluid Dynamics. Springer, New YorkCrossRefGoogle Scholar
  6. Chang RY, Wang ML (1983) Shifted Legendre direct method for variational problems. J. Optim. Theory Appl. 39(2):299–307MathSciNetCrossRefGoogle Scholar
  7. 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):147–167MathSciNetCrossRefGoogle Scholar
  8. 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):693–699MathSciNetCrossRefGoogle Scholar
  9. Defez E, Hervás A, Soler L, Tung MM (2007) Numerical solutions of matrix differential models cubic spline II. Math Comput Model 46(5–6):657–669MathSciNetCrossRefGoogle Scholar
  10. 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):865–882MathSciNetCrossRefGoogle Scholar
  11. Defez E, Tung MM, Ibáñez JJ, Sastre J (2012b) Approximating and computing nonlinear matrix differential models. Math Comput Model 55(7–8):2012–2022MathSciNetCrossRefGoogle Scholar
  12. 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):187–199Google Scholar
  13. Ezz-Eldien SS (2016) New quadrature approach based on operational matrix for solving a class of fractional variational problems. J Comput Phys 317:362–381MathSciNetCrossRefGoogle Scholar
  14. 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 MathSciNetCrossRefGoogle Scholar
  15. Ezz-Eldien SS (2018b) On solving systems of multi-pantograph equations via spectral tau method. Appl Math Comput 321:63–73MathSciNetGoogle Scholar
  16. 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 CrossRefGoogle Scholar
  17. 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):011010CrossRefGoogle Scholar
  18. 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):295–320MathSciNetCrossRefGoogle Scholar
  19. 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 57:246–263MathSciNetCrossRefGoogle Scholar
  20. Flett TM (1980) Differential analysis: differentiation, differential equations and differential inequalities. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  21. 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):1431–1440MathSciNetCrossRefGoogle Scholar
  22. 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):1377–1390MathSciNetCrossRefGoogle Scholar
  23. 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):307–318MathSciNetGoogle Scholar
  24. 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)Google Scholar
  25. Mishra VN, Mishra LN (2012) Trigonometric approximation of signals (functions) in \(L_p\)-norm. Int J Contemp Math Sci 7(19):909–918MathSciNetzbMATHGoogle Scholar
  26. 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 344:15–24MathSciNetCrossRefGoogle Scholar
  27. 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):257–264MathSciNetCrossRefGoogle Scholar
  28. 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 38:327–346MathSciNetzbMATHGoogle Scholar
  29. Rivlin TJ (1969) An introduction to the approximation of functions. Blaisdell Publishing Company, WalthamzbMATHGoogle Scholar
  30. Tang T, Xu X, Cheng J (2008) On spectral methods for Volterra type integral equations and the convergence analysis. J Comput Math 26(6):825–837MathSciNetzbMATHGoogle Scholar
  31. Tung MM, Defez E, Sastre J (2008) Numerical solutions of second-order matrix models using cubic-matrix splines. Comput Math Appl 56(10):2561–2571MathSciNetCrossRefGoogle Scholar
  32. Vandana (2017) A study of dynamic inventory involving economic ordering of commodity. Ph.D. Thesis, Pt. Ravishankar Shukla University Raipur, 492010, Chhattisgarh, IndiaGoogle Scholar
  33. 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 8:294–311CrossRefGoogle Scholar
  34. Zhao J, Xiao J, Ford NJ (2014) Collocation methods for fractional integro-differential equations with weakly singular kernels. Numer Algorithm 65(4):723–743MathSciNetCrossRefGoogle Scholar
  35. Zhou B, Cai GB, Duan GR (2013) Stabilisation of time-varying linear systems via Lyapunov differential equations. Int J Control 86(2):332–347MathSciNetCrossRefGoogle Scholar

Copyright information

© Shiraz University 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematics, Statistics and Computer SciencesSemnan UniversitySemnanIran
  2. 2.Young Researchers and Elite Club, Karaj BranchIslamic Azad UniversityKarajIran

Personalised recommendations