Numerical Solution of Second-Order Linear Fredholm Integro-Differetial Equations by Trigonometric Scaling Functions

ABSTRACT

The main aim of this paper is to apply the Hermite trigonometric scaling function on [0, 2π] which is constructed for Hermite interpolation for the linear Fredholm integro-differential equation of second order. This equation is usually difficult to solve analytically. Our approach consists of reducing the problem to a set of algebraic linear equations by expanding the approximate solution. Some numerical example is included to demonstrate the validity and applicability of the presented technique, the method produces very accurate results, and a comparison is made with exiting results. An estimation of error bound for this method is presented.

The main aim of this paper is to apply the Hermite trigonometric scaling function on [0, 2π] which is constructed for Hermite interpolation for the linear Fredholm integro-differential equation of second order. This equation is usually difficult to solve analytically. Our approach consists of reducing the problem to a set of algebraic linear equations by expanding the approximate solution. Some numerical example is included to demonstrate the validity and applicability of the presented technique, the method produces very accurate results, and a comparison is made with exiting results. An estimation of error bound for this method is presented.

KEYWORDS

Numerical Technique, Fredholm Integro-Differential Equations, Hermite Trigonometric Wavelets, Operational Matrix, Error Estimates

Numerical Technique, Fredholm Integro-Differential Equations, Hermite Trigonometric Wavelets, Operational Matrix, Error Estimates

Cite this paper

Safdari, H. and Aghdam, Y. (2015) Numerical Solution of Second-Order Linear Fredholm Integro-Differetial Equations by Trigonometric Scaling Functions.*Open Journal of Applied Sciences*, **5**, 135-144. doi: 10.4236/ojapps.2015.54014.

Safdari, H. and Aghdam, Y. (2015) Numerical Solution of Second-Order Linear Fredholm Integro-Differetial Equations by Trigonometric Scaling Functions.

References

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http://dx.doi.org/10.1016/S0885-3924(02)00393-7

[2] Wazwaz, D.D., Dimitrova, M.B. and Dishliev. A.B. (2000) Oscillation of the Bounded Solutions of Impulsive Differential-Difference Equations of Second Order. Applied Mathematics and Computation, 114, 61-68.

http://dx.doi.org/10.1016/S0096-3003(99)00102-2

[3] Gulsu, M. and Sezer, M. (2006) A Taylor Polynomial Approach for Solving Differential-Difference Equations. Journal of Computational and Applied Mathematics, 186, 349-369.

http://dx.doi.org/10.1016/j.cam.2005.02.009

[4] Maleknejad, K., Mirzaee, F. and Abbasbandy, S. (2004) Solving Linear Integro-Differential Equations System by Using Rationalized Haar Functions Method. Applied Mathematics and Computation, 155, 317-328. http://dx.doi.org/10.1016/S0096-3003(03)00778-1

[5] Maleknejad, K., Tavassoli, M. and Kajani, M. (2004) Solving Linear Integro-Differential Equation by Galerkin Methods with Hybrid Functions. Journal of Computational and Applied Mathematics, 159, 603-612. http://dx.doi.org/10.1016/j.amc.2003.10.046

[6] Ganesh, M. and Sloan, I.H. (1999) Optimal Order Spline Methods for Nonlinear Differential and Integro-Differential Equations. Applied Numerical Mathematics, 29, 445-478.

http://dx.doi.org/10.1016/S0168-9274(98)00067-1

[7] Dascoglua, A. and Sezer, M. (2005) Chebyshev Polynomial Solutions of Systems of Higher-Order Linear Fredholm- Volterra Integro-Differential Equations. Journal of the Franklin Institute, 342, 688-701. http://dx.doi.org/10.1016/j.jfranklin.2005.04.001

[8] Faour, A.L. and Saeed, R.K. (2006) Solution of a System of Linear Volterra Integral and Integro-Differential Equations by Spectral Method. Journal of Al-Nahrain University/Science, 62, 30-46.

[9] Danfu, H. and Xufeng, A.S. (2007) Numerical Solution of Integro-Differential Equations by Appling CAS Wavelet Operational Matrix of Integration. Applied Mathematics and Computation, 194, 460-466.

http://dx.doi.org/10.1016/j.amc.2007.04.048

[10] Baker, C. and Tang, A. (1997) Stability of Continuous Implicit Runge-Kutta Methods for Volterra Integro-Differential Systems with Unbounded Delays. Applied Numerical Mathematics, 24, 153-173.

http://dx.doi.org/10.1016/S0168-9274(97)00018-4

[11] El-Sayed, S., Kaya, D. and Zarea, S. (2004) The Decomposition Method Applied to Solve High Order Linear Volterra- Fredholm Integro-Differential Equations. International Journal of Nonlinear Sciences and Numerical Simulation, 52, 105-112.

[12] Sezer, M. and Gulsu, M. (2007) Polynomial Solution of the Most General Linear Fredholm-Volterra Integro Differential- Difference Equations by Means of Taylor Collocation Method. Applied Mathematics and Computation, 185, 646-657.

http://dx.doi.org/10.1016/j.amc.2006.07.051

[13] Sezer, M. and Gulsu, M. (2005) A New Polynomial Approach for Solving Difference and Fredholm Integro-Difference Equations with Mixed Argument. Applied Mathematics and Computation, 171, 332-344. http://dx.doi.org/10.1016/j.amc.2005.01.051

[14] Chui, C.K. and Mhaskar, H.N. (1993) On Trigonometric Wavelets. Constructive Approximation, 9, 167-190. http://dx.doi.org/10.1007/BF01198002

[15] Prestin, J. (2001) Trigonometric Wavelets. In: Jain, P.K., et al., Eds., Wavelet and Allied Topics, Narosa Publishing House, New Delhi, 183-217.

[16] Themistoclakis, W. (1999) Trigonometric Wavelet Interpolation in Besov Spaces. Facta Univ. (Nis) Ser. Math. Inform, 14, 49-70.

[17] Quak, E. (1996) Trigonometric Wavelets for Hermite Interpolation. Department of Mathematics, Texas A. M. University, 65, 683-722.

[18] Chen, W.S. and Lin, W. (1997) Hadamard Singular Integral Equations and Its Hermite Wavelet Methods. Proceedings of the 5th International Colloquium on Finite Dimensional Complex Analysis, Beijing, 13-22.

[19] Chen, W.S. and Lin, W. (2002) Trigonometric Hermite Wavelet and Natural Integral Equations for Stockes Problem. International Conference on Wavelet Analysis and Its Applications, Guangzhou, 73-86.

[20] Lakestani, M. and Saray, B.N. (2010) Numerical Solution of Telegraph Equation Using Interpolating Scaling Functions. Computer and Mathematics with Application, 60, 1964-1972.

http://dx.doi.org/10.1016/j.camwa.2010.07.030

[1] Desmond, R.A., Weiss, H.L., Arani, R.B., Soong, S.-J., Wood, M.J., Fiddian, P., Gnann, J. and Whitley, R.J. (2002) Clinical Applications for Change-Point Analysis of Herpes Zoster Pain. Journal of Pain and Symptom Management, 23, 510-516.

http://dx.doi.org/10.1016/S0885-3924(02)00393-7

[2] Wazwaz, D.D., Dimitrova, M.B. and Dishliev. A.B. (2000) Oscillation of the Bounded Solutions of Impulsive Differential-Difference Equations of Second Order. Applied Mathematics and Computation, 114, 61-68.

http://dx.doi.org/10.1016/S0096-3003(99)00102-2

[3] Gulsu, M. and Sezer, M. (2006) A Taylor Polynomial Approach for Solving Differential-Difference Equations. Journal of Computational and Applied Mathematics, 186, 349-369.

http://dx.doi.org/10.1016/j.cam.2005.02.009

[4] Maleknejad, K., Mirzaee, F. and Abbasbandy, S. (2004) Solving Linear Integro-Differential Equations System by Using Rationalized Haar Functions Method. Applied Mathematics and Computation, 155, 317-328. http://dx.doi.org/10.1016/S0096-3003(03)00778-1

[5] Maleknejad, K., Tavassoli, M. and Kajani, M. (2004) Solving Linear Integro-Differential Equation by Galerkin Methods with Hybrid Functions. Journal of Computational and Applied Mathematics, 159, 603-612. http://dx.doi.org/10.1016/j.amc.2003.10.046

[6] Ganesh, M. and Sloan, I.H. (1999) Optimal Order Spline Methods for Nonlinear Differential and Integro-Differential Equations. Applied Numerical Mathematics, 29, 445-478.

http://dx.doi.org/10.1016/S0168-9274(98)00067-1

[7] Dascoglua, A. and Sezer, M. (2005) Chebyshev Polynomial Solutions of Systems of Higher-Order Linear Fredholm- Volterra Integro-Differential Equations. Journal of the Franklin Institute, 342, 688-701. http://dx.doi.org/10.1016/j.jfranklin.2005.04.001

[8] Faour, A.L. and Saeed, R.K. (2006) Solution of a System of Linear Volterra Integral and Integro-Differential Equations by Spectral Method. Journal of Al-Nahrain University/Science, 62, 30-46.

[9] Danfu, H. and Xufeng, A.S. (2007) Numerical Solution of Integro-Differential Equations by Appling CAS Wavelet Operational Matrix of Integration. Applied Mathematics and Computation, 194, 460-466.

http://dx.doi.org/10.1016/j.amc.2007.04.048

[10] Baker, C. and Tang, A. (1997) Stability of Continuous Implicit Runge-Kutta Methods for Volterra Integro-Differential Systems with Unbounded Delays. Applied Numerical Mathematics, 24, 153-173.

http://dx.doi.org/10.1016/S0168-9274(97)00018-4

[11] El-Sayed, S., Kaya, D. and Zarea, S. (2004) The Decomposition Method Applied to Solve High Order Linear Volterra- Fredholm Integro-Differential Equations. International Journal of Nonlinear Sciences and Numerical Simulation, 52, 105-112.

[12] Sezer, M. and Gulsu, M. (2007) Polynomial Solution of the Most General Linear Fredholm-Volterra Integro Differential- Difference Equations by Means of Taylor Collocation Method. Applied Mathematics and Computation, 185, 646-657.

http://dx.doi.org/10.1016/j.amc.2006.07.051

[13] Sezer, M. and Gulsu, M. (2005) A New Polynomial Approach for Solving Difference and Fredholm Integro-Difference Equations with Mixed Argument. Applied Mathematics and Computation, 171, 332-344. http://dx.doi.org/10.1016/j.amc.2005.01.051

[14] Chui, C.K. and Mhaskar, H.N. (1993) On Trigonometric Wavelets. Constructive Approximation, 9, 167-190. http://dx.doi.org/10.1007/BF01198002

[15] Prestin, J. (2001) Trigonometric Wavelets. In: Jain, P.K., et al., Eds., Wavelet and Allied Topics, Narosa Publishing House, New Delhi, 183-217.

[16] Themistoclakis, W. (1999) Trigonometric Wavelet Interpolation in Besov Spaces. Facta Univ. (Nis) Ser. Math. Inform, 14, 49-70.

[17] Quak, E. (1996) Trigonometric Wavelets for Hermite Interpolation. Department of Mathematics, Texas A. M. University, 65, 683-722.

[18] Chen, W.S. and Lin, W. (1997) Hadamard Singular Integral Equations and Its Hermite Wavelet Methods. Proceedings of the 5th International Colloquium on Finite Dimensional Complex Analysis, Beijing, 13-22.

[19] Chen, W.S. and Lin, W. (2002) Trigonometric Hermite Wavelet and Natural Integral Equations for Stockes Problem. International Conference on Wavelet Analysis and Its Applications, Guangzhou, 73-86.

[20] Lakestani, M. and Saray, B.N. (2010) Numerical Solution of Telegraph Equation Using Interpolating Scaling Functions. Computer and Mathematics with Application, 60, 1964-1972.

http://dx.doi.org/10.1016/j.camwa.2010.07.030