Numerical Solution of Nonlinear Integro-Differential Equations with Initial Conditions by Bernstein Operational Matrix of Derivative

ABSTRACT

In this paper, we present a practical matrix method for solving nonlinear Volterra-Fredholm integro-differential equations under initial conditions in terms of Bernstein polynomials on the interval [0,1]. The nonlinear part is approximated in the form of matrices’ equations by operational matrices of Bernstein polynomials, and the differential part is approximated in the form of matrices’ equations by derivative operational matrix of Bernstein polynomials. Finally, the main equation is transformed into a nonlinear equations system, and the unknown of the main equation is then approximated. We also give some numerical examples to show the applicability of the operational matrices for solving nonlinear Volterra-Fredholm integro-differential equations (NVFIDEs).

Cite this paper

B. Basirat and M. Shahdadi, "Numerical Solution of Nonlinear Integro-Differential Equations with Initial Conditions by Bernstein Operational Matrix of Derivative,"*International Journal of Modern Nonlinear Theory and Application*, Vol. 2 No. 2, 2013, pp. 141-149. doi: 10.4236/ijmnta.2013.22018.

B. Basirat and M. Shahdadi, "Numerical Solution of Nonlinear Integro-Differential Equations with Initial Conditions by Bernstein Operational Matrix of Derivative,"

References

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[2] F. Bloom, “Asymptotic Bounds for Solutions to a System of Damped Integro-Differential Equations of Electromagnetic Theory,” Journal of Mathematical Analysis and Applications, Vol. 73, 1980, pp. 524-542.

[3] M. A. Jaswon and G. T. Symm, “Integral Equation Methods in Potential Theory and Elastostatics,” Academic Press, London, 1977.

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[5] P. Schiavane and C. Constanda and A. Mioduchowski, “Integral Methods in Science and Engineering,” Birkhauser, Boston, 2002.

[6] B. I. Smetanin, “On an Integral Equation for Axially-Symmetric Problems in the Case of an Elastic Body Containing an Inclusion,” Journal of Applied Mathematics and Mechanics, Vol. 55, No. 3, 1991, pp. 371-375.

[7] N. N. Voitovich and O. O. Reshnyak, “Solutions of Nonlinear Integral Equation of Synthesis of the Linear Antenna Arrays,” BSUAE Journal of Applied Electromagnetism, Vol. 2, No. 1, 1999, pp. 43-52.

[8] L. M. Delves and J. Walsh, “Numerical Solution of Integral Equation,” Oxford University Press, London, 1974.

[9] G. Micula and P. Pavel, “Differential and Integral Equations through Practical Problems and Exercises,” Kluwer, 1992.

[10] R. K. Miller, “Nonlinear Volterra Integral Equations,” Menlo Park, 1967.

[11] E. H. Doha, A. H. Bhrawy and M. A. Saker, “Integrals of Bernestein Polynomials: An Application for the Solution of High Even-Order Differential Equations,” Applied Mathematics Letters, Vol. 24, No. 4, 2011, pp. 559-565.

[12] R. T. Farouki and V. T. Rajan, “Algorithms for Polynomials in Bernstein Form,” Computer Aided Geometric Design, Vol. 5, No. 1, 1988, pp. 1-26.

[13] K. Maleknejad, E. Hashemizadeh and B. Basirat, “Computational Method Based on Bernestein Operational Matrices for Nonlinear Volterra—Fredholm-Hammerstein Integral Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 17, No. 1, 2012, pp. 52-61. doi:10.1016/j.cnsns.2011.04.023

[14] S. A. Yousefi and M. Behroozifar, “Operational Matrices of Bernstein Polynomials and Their Applications,” International Journal of Systems Science, Vol. 41, No. 6, 2010, pp. 709-716. doi:10.1080/00207720903154783

[15] Z. Avazzadeh, M. Heydari and G. B. Loghmani, “Numerical Solution of Fredholm Integral Equations of the Second Kind by Using Integral Mean Value Theorem,” Applied Mathematical Modelling, Vol. 35, No. 5, 2011, pp. 2374-2383.

[16] F. Mirzaee, “The RHFs for Solution of Nonlinear Fredholm Integro-Differential Equations,” Applied Mathematical Modelling, Vol. 5, No. 70, 2011, pp. 3453-3464.

[17] W. Wang, “An Algorithm for Solving the High-Order Nonlinear Volterra-Fredholm Integro-Differential Equation with Mechanization,” Applied Mathematics and Computation, Vol. 172, No. 1, 2006, pp. 1-23.

[1] M. A. Abdou, “On Asymptotic Methods for Fredholm-Volterra Integral Equation of the Second Kind in Contact Problems,” Journal of Computational and Applied Mathematics, Vol. 154, No. 2, 2003, pp. 431-446.

[2] F. Bloom, “Asymptotic Bounds for Solutions to a System of Damped Integro-Differential Equations of Electromagnetic Theory,” Journal of Mathematical Analysis and Applications, Vol. 73, 1980, pp. 524-542.

[3] M. A. Jaswon and G. T. Symm, “Integral Equation Methods in Potential Theory and Elastostatics,” Academic Press, London, 1977.

[4] S. Jiang and V. Rokhlin, “Second Kind Integral Equations for the Classical Potential Theory on Open Surface II,” Journal of Computational Physics, Vol. 195, No. 3, 2004, pp. 1-16.

[5] P. Schiavane and C. Constanda and A. Mioduchowski, “Integral Methods in Science and Engineering,” Birkhauser, Boston, 2002.

[6] B. I. Smetanin, “On an Integral Equation for Axially-Symmetric Problems in the Case of an Elastic Body Containing an Inclusion,” Journal of Applied Mathematics and Mechanics, Vol. 55, No. 3, 1991, pp. 371-375.

[7] N. N. Voitovich and O. O. Reshnyak, “Solutions of Nonlinear Integral Equation of Synthesis of the Linear Antenna Arrays,” BSUAE Journal of Applied Electromagnetism, Vol. 2, No. 1, 1999, pp. 43-52.

[8] L. M. Delves and J. Walsh, “Numerical Solution of Integral Equation,” Oxford University Press, London, 1974.

[9] G. Micula and P. Pavel, “Differential and Integral Equations through Practical Problems and Exercises,” Kluwer, 1992.

[10] R. K. Miller, “Nonlinear Volterra Integral Equations,” Menlo Park, 1967.

[11] E. H. Doha, A. H. Bhrawy and M. A. Saker, “Integrals of Bernestein Polynomials: An Application for the Solution of High Even-Order Differential Equations,” Applied Mathematics Letters, Vol. 24, No. 4, 2011, pp. 559-565.

[12] R. T. Farouki and V. T. Rajan, “Algorithms for Polynomials in Bernstein Form,” Computer Aided Geometric Design, Vol. 5, No. 1, 1988, pp. 1-26.

[13] K. Maleknejad, E. Hashemizadeh and B. Basirat, “Computational Method Based on Bernestein Operational Matrices for Nonlinear Volterra—Fredholm-Hammerstein Integral Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 17, No. 1, 2012, pp. 52-61. doi:10.1016/j.cnsns.2011.04.023

[14] S. A. Yousefi and M. Behroozifar, “Operational Matrices of Bernstein Polynomials and Their Applications,” International Journal of Systems Science, Vol. 41, No. 6, 2010, pp. 709-716. doi:10.1080/00207720903154783

[15] Z. Avazzadeh, M. Heydari and G. B. Loghmani, “Numerical Solution of Fredholm Integral Equations of the Second Kind by Using Integral Mean Value Theorem,” Applied Mathematical Modelling, Vol. 35, No. 5, 2011, pp. 2374-2383.

[16] F. Mirzaee, “The RHFs for Solution of Nonlinear Fredholm Integro-Differential Equations,” Applied Mathematical Modelling, Vol. 5, No. 70, 2011, pp. 3453-3464.

[17] W. Wang, “An Algorithm for Solving the High-Order Nonlinear Volterra-Fredholm Integro-Differential Equation with Mechanization,” Applied Mathematics and Computation, Vol. 172, No. 1, 2006, pp. 1-23.