Solving Systems of Volterra Integral Equations with Cardinal Splines


This work is a continuation of the earlier article [1]. We establish new numerical methods for solving systems of Volterra integral equations with cardinal splines. The unknown functions are expressed as a linear combination of horizontal translations of certain cardinal spline functions with small compact supports. Then a simple system of equations on the coefficients is acquired for the system of integral equations. It is relatively straight forward to solve the system of unknowns and an approximation of the original solution with high accuracy is achieved. Several cardinal splines are applied in the paper to enhance the accuracy. The sufficient condition for the existence of the inverse matrix is examined and the convergence rate is investigated. We demonstrated the value of the methods using several examples.

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Liu, X. , Liu, Z. and Xie, J. (2015) Solving Systems of Volterra Integral Equations with Cardinal Splines. Journal of Applied Mathematics and Physics, 3, 1422-1430. doi: 10.4236/jamp.2015.311170.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Liu, X. and Xie, J. (2014) Numerical Methods for Solving Systems of Fredholm Integral Equations with Cardinal Splines. AIP Conference Proceedings, 1637, 590.
[2] Adawi, A. and Awawdeh, F. (2009) A Numerical Method for Solving Linear Integral Equations. International Journal of Contemporary Mathematical Sciences, 4, 485-496.
[3] Polyanin, A.D. (1998) Handbook of Integral Equations. CRC Press LLC, Boca Raton.
[4] Saeed, R.K. and Ahmed, C.S. (2008) Approximate Solution for the System of Non-Linear Volterra Integral Equations of the Second Kind by Using Block-by-block Method. Australian Journal of Basic and Applied Sciences, 2, 114-124.
[5] Schoenberg, I.J. (1964) On Trigonometric Spline Functions. Journal of Mathematics and Mechanics, 13, 795-825.
[6] Chui, C.K. (1988) Multivariate Splines. SIAM, Philadelphia.
[7] Liu, X. (2001) Bivariate Cardinal Spline Functions for Digital Signal Processing. In: Kopotum, K., Lyche, T. and Neamtu, M., Eds., Trends in Approximation Theory, Vanderbilt University, Nashville, 261-271.
[8] Liu, X. (2007) Interpolation by Cardinal Exponential Splines. The Journal of Information and Computational Science, 4, 179-194.
[9] Liu, X. (2013) The Applications of Orthonormal and Cardinal Splines in Solving Linear Integral Equations. In: Akis, V., Ed., Essays on Mathematics and Statistics, V4, Athens Institute for Education and Research, 41-58.
[10] Liu, X., Xie, J. and Xu, L. (2014) The Applications of Cardinal Trigonometric Splines in Solving Nonlinear Integral Equations. Applied Mathematics, 2014, Article ID: 213909.
[11] Liu, X. (2006) Univariate and Bivariate Orthornormal Splines and Cardinal Splines on the Compact Supports. Journal of Computational and Applied Mathematics, 195, 93-105.

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