# SOLVING VOLTERRA-FREDHOLM INTEGRAL EQUATIONS BY QUADRATIC SPLINE FUNCTION

• Sarfraz Hassan Salim Department of Mathematics, College of Science, Salahaddin University-Erbil, Iraq
• Rostam Karim Saeed Department of Mathematics, College of Science, Salahaddin University-Erbil, Iraq
• Karwan Hama Faraj Jwamer Department of Mathematics, College of Science, Salahaddin University-Erbil, Iraq
Keywords: Volterra Integral Equation, Fredholm Integral Equation, Spline Function

### Abstract

Using the quadratic spline function, this paper finds the numerical solution of mixed Volterra-Fredholm integral equations of the second kind. The proposed method is based on employing the quadratic spline function of the unknown function at an arbitrary point and using the integration method to turn the Volterra-Fredholm integral equation into a system of linear equations with respect to the unknown function. An approximate solution can be easily established by solving the given system. This is accomplished with the help of a computer program that runs on Python 3.9..

### References

 J. T. A. Al-Miah and A. H. S. Taie, A new Method for Solutions Volterra-Fredholm Integral Equation of the Second Kind, IOP Conf. Series: Journal of Physics: Conf. Series, 1294 (2019) 032026.
 K. E. Atkinson, The numerical solution of integral equation of the second kind, 4, Cambridge university press, 1997.,

 J. E.Bekelman, Y. Ly, and C. P. Gross, Scope and impact of financial conflicts of interest in biomedical research: a systematic review, JAMA, 289(2003), No. 19, 454–465.

 Cheney, W.C and D.Kincaid, Numerical Mathematics and Computing, Brooks/Cole Publication Company, (1999). I. Podlubny, Fractional Differential Equations. San Diego: Elsevier, 1999.
 H. L. Dastjerdi and F. M. M. Ghaini, Numerical solution of Volterra–Fredholm integral equations by moving least square method and Chebyshev polynomials, Applied Mathematical Modelling, 36(2012), 3283-–3288.
 M. Didgara and A. Vahidi, Approximate Solution of Linear Volterra-Fredholm Integral Equations and Systems of Volterra-Fredholm Integral Equations using Taylor Expansion Method, Iranian Journal of Mathematical Sciences and Informatics, 15(2020), No.1, 31-50.
 P. M. A. Hasan and N. A. Sulaiman, Numerical Treatment of Mixed Volterra-Fredholm Integral Equations Using Trigonometric Functions and Laguerre Polynomials, ZANCO Journal of Pure and Applied Sciences, 30(2016), No.6, 97–106..
 A. J. Jerry, Introduction to Integral Equation with Application, Marcel Dekker, 1985.
 T. Kaminaka and M. Wadati, Higher order solutions of Lieb-Liniger integral equation, Physics Letters A, 375(2011), No.24, 2460–2464.
 E. G. Ladopoulos, Reserves exploration by real-time expert seismology and non linear singular integral equations, Oil Gas and Coal Technol., 5(2011), No.4, 299–315.
 W. A. Lange and J. M. Herbert, Symmetric versus asymmetric discretization of the integral equations in polarizablecontinuum solvation models, Chemical Physics Letters, 509 (2011), No.1, 77-87.
 S. Micula, On Some Iterative Numerical Methods for Mixed Volterra–Fredholm Integral Equations, Symmetry, 11 (2019), 1200; doi:10.3390/sym11101200,10 pages.
 R. K. Saeed and K. A. Berdawood, Solving Two-dimensional Linear Volterra-Fredholm Integral Equations of the Second Kind by Using Successive Approximation Method and Method of Successive Substitutions, ZANCO Journal of Pure and Applied Sciences, 28 (2016), No.2, 35–46.
 K. Y. Wang and Q. S. Wang, Lagrange collocation method for solving Volterra-Fredholm integral equations, Applied Mathematics and Computation, 219 (2013), No.21, 10434–10440.
 S. Yalcinbas, Taylor polynomial solution of nonlinear Volterra-Fredholm integral equations, Applied Mathematics and Computation, 127 (2002), No.2-3, 195–206.
 S. Yalcinbas and M.Seser, The approximation solution of high-order linear volterra-fredholm integro-differential equations in terms of Taylor polynomial, Applied Mathematics and Computation, 112 (2000), No.2-3, 291-–308.
 A. D. Michal, Integral Equations and Functionals, Mathematics Magazin, 24(1950), 83-95.
 R. S. Anderssen, F. R. De Hoog and M. A. Lukas, The application and numerical solution of integral equations, Sijthoff & Noordoff International Publishers B.v., Alphenan den Rijn, the Netherlands, 1980.
 C. Corduneanu, Integral Equations and applications, Cambridge University Press , United Kinggom(1991).
 M. Rahman, Aapplied differential equations for scientists and engineers, Ordinary differential equations, Southamptom: Computational Mechanics Publication,(1991).
 M. Rahman, Integral equations and their applications, WIT press, Southampton, Boston,(2007).
 R. K. Saeed, K. H. F. Jwamer, and F. K. Hamasalh, Introduction to Numerical Analysis, University of Sulaimani, 28 (2015).
 S. H. Salim, R. K. Saeed and K.H.F. Jwamer, Solving Volterra-Fredholm integral equations by linear spline function, Glob. Stoch. Anal., (accepted) (2022).  A. D. Michal, Integral Equations and Functionals, Mathematics Magazin, 24(1950), 83-95.
 S. S. Ahmed, Numerical Solutions of Linear Volterra Integro-Differential Equations. MSc thesis. University of Technology, Iraq, 2002.
 L.M. Delves and, J. Walsh, Numerical solution of integral equations, Clarednaom press, Oxford, 1974.
Published
2022-12-02
How to Cite
Salim, S., Saeed, R., & Faraj Jwamer, K. (2022). SOLVING VOLTERRA-FREDHOLM INTEGRAL EQUATIONS BY QUADRATIC SPLINE FUNCTION. Journal of Al-Qadisiyah for Computer Science and Mathematics, 14(4), Math Page 10-19. https://doi.org/10.29304/jqcm.2022.14.4.1092
Section
Math Articles