A Computational Method for Volterra Integro-Differential Equation

Abstract

Bu çalışmada, lineer birinci mertebeden Volterra integro-diferansiyel denklem içeren başlangıç değer problemini ele almaktayız. Bu problemi nümerik olarak çözmek için yeni bir sonlu fark metodu veriyoruz. Bu metot, kalan terimi integral biçiminde olan interpolasyon quadratür formülleri ve üstel baz fonksiyonunu içeren integral özdeşliklerinden meydana gelmektedir. Ayrıca, bu metodun hata analizinin bir sonucu olarak, ayrık maksimum normda birinci mertebeden yakınsaklığı ispatlandı. Son olarak, elde edilen teorik sonuçları destekleyen nümerik örnek verildi.

Keywords

• Başlangıç-Değer Problemi,Hata Değerlendirmesi,Sonlu Fark Metodu,Volterra İntegro-Diferansiyel Denklem

A Computational Method for Volterra Integro-Differential Equation

Abstract

In this paper, we examine the initial value problem for a linear first order Volterra integro-differential equation. In order to solve the problem computationally, we present a novel finite difference method, which is based on the method of integral identities with the use of the basis functions and interpolating quadrature rules with remainder term in integral form. Furthermore, as a consequence of error analysis the method is proved to be first-order convergent in the discrete maximum norm. Finally, an example is provided to support our theoretical results.

Keywords

• Error Estimate,Finite Difference Method,Initial Value Problem,Volterra Integro-Differential Equation

References

1. Amiraliyev, G.M., Mamedov, Y.D. 1995. Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations. Turkish Journal of Mathematics, 19, 207-222.
2. Amiraliyev, G.M., Sevgin, S. 2006. Uniform difference method for singularly perturbed Volterra integro-differential equations. Applied Mathematics and Computation, 179(2), 731-741.
3. Amiraliyev, G.M., Yilmaz, B. 2014. Fitted difference method for a singularly perturbed initial value problem. International Journal of Mathematics and Computation, 22(1), 1-10.
4. Babolian, E., Shamloo, A.S. 2008. Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions. Journal of Computational and Applied Mathematics, 214(2), 495-508.
5. Burton, T.A. 2005. Volterra Integral and Differential Equations 2nd ed, Elsevier, Amsterdam.
6. Chang, S.H. 1982. On certain extrapolation methods for the numerical solution of integro-differential equations. Mathematics of Computation, 39(159), 165-171.
7. De Gaetano, A., Arino, O. 2000. Mathematical modelling of the intravenous glucose tolerance test. Journal of Mathematical Biology, 40(2), 136-168.
8. Fazeli, S., Hojjati, G. 2015. Numerical solution of Volterra integro-differential equations by superimplicit multistep collocation methods. Numerical Algorithms, 68(4), 741-768.

9. Hackbusch, W. 1995. Integral Equations Theory and Numerical Treatment, Birkhauser Verlag, Basel.
10. Hoppensteadt, F.C., Jackiewicz, Z., Kowal, B.Z. 2007. Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels. BIT Numerical Mathematics, 47(2), 325-350.
11. Jerri, A. 1999. Introduction to Integral Equations with Applications, Wiley, New York.
12. Khater, A.H., Shamardan A.B., Callebaut D.K., Sakran, M.R.A. 2007. Numerical solutions of integral and integro-differential equations using Legendre polynomials. Numerical Algorithms, 46(3), 195-218.
13. Kudu, M., Amirali, I., Amiraliyev, G.M. 2016. A finite-difference method for a singularly perturbed delay integro-differential equation. Journal of Computational and Applied Mathematics, 308, 379-390.
14. Kythe, P.K., Puri, P. 2002. Computational Methods for Linear Integral Equations, Springer, New York.
15. Lakshmikantham, V., Rao, M.R.M. 1995. Theory of Integro-Differential Equations, Gordon and Breach Science Publishers, Amsterdam.
16. Mehdiyeva, G., Imanova, M., Ibrahimov, V. 2011. Application of the hybrid methods to solving Volterra integro-differential equations. World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences, 5(5), 788-792.
17. Okayama, T. 2018. Theoretical analysis of a Sinc-Nyström method for Volterra integro-differential equations and its improvement. Applied Mathematics and Computation, 324, 1-15.
18. Rahman, M. 2007. Integral Equations and their Applications, WIT Press, Boston.
19. Song, Y., Baker, C.T.H. 2004. Qualitative behaviour of numerical approximations to Volterra integro-differential equations, Journal of Computational and Applied Mathematics, 172(1), 101-115.
20. Volterra, V. 1959. Theory of Functionals and of Integral and Integro-differential Equations, Dover Publications, New York, 138-146. Wazwaz, A.M. 2011. Linear and Nonlinear Integral Equations Methods and Applications, Springer-Verlag, Berlin.