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Numerical Solutions of Matrix Differential Models Using Higher-Order Matrix Splines

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Abstract

This paper deals with the construction of approximate solution of first-order matrix linear differential equations using higher-order matrix splines. An estimation of the approximation error, an algorithm for its implementation and some illustrative examples are included.

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References

  1. Al-Said E.A., Noor M.A.: Cubic splines method for a system of third-order boundary value problems. Appl. Math. Comput. 142, 195–204 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ascher U., Mattheij R., Russell R.: Numerical solutions of boundary value problems for ordinary differential equations. Prentice Hall, New Jersey, USA (1988)

    Google Scholar 

  3. Barnett S.: Matrices in Control Theory. Van Nostrand, Reinhold (1971)

    MATH  Google Scholar 

  4. Blanes S., Casas F., Oteo J.A., Ros J.: Magnus and Fer expansion for matrix differential equations: the convergence problem. J. Phys. Appl. 31, 259–268 (1998)

    MathSciNet  MATH  Google Scholar 

  5. Boggs P.T.: The solution of nonlinear systems of equations by a-stable integration techniques. SIAM J. Numer. Anal. 8(4), 767–785 (1971)

    Article  MathSciNet  Google Scholar 

  6. Defez E., Hervás A., Law A., Villanueva-Oller J., Villanueva R.: Matrixcubic splines for progressive transmission of images. J. Math. Imaging Vision 17(1), 41–53 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Defez E., Soler L., Hervás A., Santamaría C.: Numerical solutions of matrix differential models using cubic matrix splines. Comput. Math. Appl. 50, 693–699 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Defez E., Soler L., Hervás A., Tung M.M.: Numerical solutions of matrix differential models using cubic matrix splines II. Mathematical and Computer Modelling 46, 657–669 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mazzia F., Trigiante A.S., Trigiante A.S.: B-spline linear multistep methods and their conitinuous extensions. SIAM J. Numer. Anal. 44(5), 1954–1973 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Faddeyev L.D.: The inverse problem in the quantum theory of scattering. J. Math. Physics 4(1), 72–104 (1963)

    Article  MathSciNet  Google Scholar 

  11. Flett, T.M.: Differential Analysis. Cambridge University Press (1980)

  12. Golub G.H., Loan C.F.V.: Matrix Computations, second edn. The Johns Hopkins University Press, Baltimore, MD, USA (1989)

    Google Scholar 

  13. Graham A.: Kronecker products and matrix calculus with applications. John Wiley & Sons, New York, USA (1981)

    MATH  Google Scholar 

  14. Jódar L., Cortés J.C.: Rational matrix approximation with a priori error bounds for non-symmetric matrix riccati equations with analytic coefficients. IMA J. Numer. Anal. 18(4), 545–561 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jódar L., Cortés J.C., Morera J.L.: Construction and computation of variable coefficient sylvester differential problems. Computers Maths. Appl. 32(8), 41–50 (1996)

    Article  MATH  Google Scholar 

  16. Jódar, L., Ponsoda, E.: Continuous numerical solutions and error bounds for matrix differential equations. In: Int. Proc. First Int. Colloq. Num. Anal., pp. 73–88. VSP, Utrecht, The Netherlands (1993)

  17. Jódar L., Ponsoda E.: Non-autonomous riccati-type matrix differential equations: Existence interval, construction of continuous numerical solutions and error bounds. IMA J. Numer. Anal. 15(1), 61–74 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Loscalzo F.R., Talbot T.D.: Spline function approximations for solutions of ordinary differential equations. SIAM J. Numer. Anal. 4(3), 433–445 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  19. Marzulli P.: Global error estimates for the standard parallel shooting method. J. Comput. Appl. Math. 34, 233–241 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Micula G., Revnic A.: An implicit numerical spline method for systems for ode’s. Appl. Math. Comput. 111, 121–132 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Reid, W.T.: Riccati Differential Equations. Academic Press (1972)

  22. Rektorys, K.: The method of discretization in time and partial differential equations. D. Reidel Pub. Co., Dordrecht (1982)

  23. Scott, M.: Invariant imbedding and its Applications to Ordinary Differential Equations. Addison-Wesley (1973)

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Correspondence to Antonio Hervás.

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Defez, E., Hervás, A., Ibáñez, J. et al. Numerical Solutions of Matrix Differential Models Using Higher-Order Matrix Splines. Mediterr. J. Math. 9, 865–882 (2012). https://doi.org/10.1007/s00009-011-0159-z

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  • DOI: https://doi.org/10.1007/s00009-011-0159-z

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