AUTHORS: I. G. Burova, N. S. Domnin, A. E. Vezhlev, A. V. Lebedeva, A. N. Pakulina

Download as PDF

The present paper is devoted to the application of local polynomial integro-differential splines to the solution of integral equations, in particular, to the solution of the integral equations of Fredholm of the second kind. To solve the Fredholm equation of the second kind, we apply local polynomial integro-differential splines of the second and third order of approximation. To calculate the integral in the formulae of a piecewise quadratic integro-differential spline and piecewise linear integro-differential spline, we propose the corresponding quadrature formula. The results of the numerical experiments are given.

KEYWORDS: polynomial splines, polynomial integro-differential splines, Fredholm equation

REFERENCES:

[1] G.Zeng, C.Chen, L.Lei X.Xu, A Modified Collocation Method for Weakly Singular Fredholm Integral Equations of Second Kind, Journal of Computational Analysis and Applications, Vol.27 No.7, 2019, pp. 1091- 1102.

[2] B.L.Panigrahi, M. Mandal, G. Nelakanti, Legendre Multi-Galerkin Methods for Fredholm Integral Equations with Weakly Singular Kernel and the Corresponding Eigenvalue Problem, Journal of Computational and Applied Mathematics, Vol.346, 2019, pp. 224-236.

[3] J.Xie, M.Yi, Numerical Research of Nonlinear System of Fractional Volterra-Fredholm Integral-Differential Equations via Block-Pulse Functions and Error Analysis, Journal of Computational and Applied Mathematics, Vol.345, 2019, pp. 159-167.

[4] A.Babaaghaie, K.Maleknejad, A New Approach for Numerical Solution of TwoDimensional Nonlinear Fredholm Integral Equations in the Most General Kind of Kernel, Based on Bernstein Polynomials and its Convergence Analysis, Journal of Computational and Applied Mathematics, Vol.344, 2018, pp. 482-494.

[5] K.Maleknejad, J.Rashidinia, T.Eftekhari, Numerical Solution of Three-Dimensional Volterra-Fredholm Integral Equations of the First and Second Kinds Based on Bernstein’s Approximation, Applied Mathematics and Computation, Vol.339, 2018, pp. 272-285.

[6] N.Negarchi, K.Nouri, Numerical Solution of Volterra-Fredholm Integral Equations Using the Collocation Method Based on a Special Form of the Mёuntz-Legendre Polynomials, Journal of Computational and Applied Mathematics, Vol.344, 2018, pp. 15-24.

[7] A.M.Rocha, J.S.Azevedo, S.P.Oliveira, M.R. Correa, Numerical Analysis of a Collocation Method for Functional Integral Equations, Applied Numerical Mathematics, Vol.134, 2018, pp. 31-45.

[8] F.Mohammadi, J.A.T.Machado, A Comparative Study of Integer and Noninteger Order Wavelets for Fractional Nonlinear Fredholm Integro-Differential Equations, Journal of Computational and Nonlinear Dynamics, Vol.13, No 8, 2018.

[9] D.Barrera, F.Elmokhtari, D.Sbibih, Two Methods Based on Bivariate Spline QuasiInterpolants for Solving Fredholm Integral Equations, Applied Numerical Mathematics, Vol.127, 2018, pp. 78-94.

[10] M.Erfanian, H.Zeidabadi, Solving of Nonlinear Fredholm Integrodifferential Equation in a Complex Plane with Rationalized Haar Wavelet Bases, Asian-European Journal of Mathematics, 2018. Article in Press. C.Allouch, P.Sablonni`ere, Iteration Methods for Fredholm Integral Equations of the Second Kind Based on Spline Quasi-Interpolants, Mathematics and Computers in Simulation, Vol. 99, 2014, pp. 19-27.

[11] I.G.Burova, O.V.Rodnikova, Integro-Differential Polynomial and Trigonometrical Splines and Quadrature Formulae, WSEAS Transactions on Mathematics, Vol.16, 2017, pp.11-18.

[12] I.G.Burova, A.G.Doronina, I.D.Miroshnichenko,A Comparison of Approximations with Left, Right and Middle Integro-Differential Polynomial Splines of the Fifth Order, WSEAS Transactions on Mathematics, Vol.16, 2017, pp. 339-349.

[13] I.G.Burova, S.V.Poluyanov, On Approximations by Polynomial And Trigonometrical Integro-Differential Splines, International Journal of Mathematical Models and Methods in Applied Sciences, Vol.10, 2016, pp. 190-199.

[14] I.G.Burova, A.G.Doronina, On Approximations by Polynomial and Nonpolynomial Integro-Differential Splines. Applied Mathematical Sciences, Vol.10, No 13- 16, 2016, pp. 735-745.

[15] I.G.Burova, On Left Integro-Differential Splines and Cauchy Problem. International Journal of Mathematical Models and Methods in Applied Sciences, Vol.9, 2015, pp. 683-690.

[16] I.G. Burova, O.V.Rodnikova, Application of Integro-differential Splines to Solving an Interpolation Problem, Computational Mathematics and Mathematical Physics, Vol.54, No 12, 2014, pp. 1903-1914.

[17] I.G.Burova, S.V.Poluyanov, Construction of Meansquare Approximation with IntegroDifferential Splines of Fifth Order and First Level, Vestnik St. Petersburg University, Mathematics, Vol.47, No 2, 2014, pp. 57-63.

[18] I.G.Burova, T.O.Evdokimova, On Construction Third Order Approximation Using Values of Integrals, WSEAS Transactions on Mathematics, Vol.13, 2014, pp. 676-683.

[19] F.-G.Lang, X.-P.Xu, On a New Approximation Method with Four Times Continuously Differentiable Quintic Integro-Differential Spline, AENG International Journal of Applied Mathematics, Vol. 47, No 4, 1, pp. 371-380.

[20] A.Bellour, D. Sbibih, A.Zidna, Two Cubic Spline Methods for Solving Fredholm Integral Equations, Applied Mathematics and Computation, Vol.276, 2016, pp. 1-11.

[21] Fengmin Chen, Patricia J. Y.Wong, Discrete Biquintic Spline Method for Fredholm Integral Equations of the Second Kind, 12th International Conference on Control, Automation, Robotics & Vision Guangzhou, China, 5-7th December 2012 (ICARCV 2012)

[22] N.Ebrahimi, J.Rashidinia, Spline Collocation for Solving System of Fredholm and Volterra Integral Equations, International Journal of Mathematical and Computational Sciences, Vol.8, No 6, 2014.

[23] P.Kalyani, P.S.Ramachandra Rao, Numerical Solution of Heat Equation through Double Interpolation, IOSR Journal of Mathematics (IOSR-JM), Vol.6, No 6, May. - Jun. 2013, pp. 58-62.

[24] Chafik Allouch, Paul Sablonnière, Driss Sbibih, Solving Fredholm Integral Equations by Approximating Kernels by Spline QuasiInterpolants, Numer Algor, Vol.56, 2011, pp. 437-453.

[25] S.Saha Ray, P.K.Sahu, Application of Semiorthogonal B-Spline Wavelets for the Solutions of Linear Second Kind Fredholm Integral Equations, Appl. Math. Inf. Sci., Vol.8 No 3, 2014, pp.1179-1184.

[26] J.Rashidinia, E.Babolian, Z.Mahmoodi, Spline Collocation for Fredholm Integral Equations, Mathematical Sciences, 5 (2), 2011,pp. 147- 158.

[27] P.S. Rama Chandra Rao, Solution of Fourth Order of Boundary Value Problems Using Spline Functions, Indian Journal of Mathematics and Mathematical Sciences, Vol. 2, No 1, 2006, pp. 47-56.

[28] P.S.Rama Chandra Rao, Solution of a Class of Boundary Value Problems using Numerical Integration, Indian journal of mathematics and mathematical sciences, Vol.2, No 2, 2006, pp. 137-146.

[29] A.S.V. Ravikanth, Numerical Treatment of Singular Boundary Value Problems, Ph.D. Thesis, National Institute of Technology, Warangal, India, 2002.

[30] J.H. Ahlberg, E. N. Nilson, J. L. Walsh,: Theory of Splines and their Applications, Mathematics in Science and Engineering, Chapt. IV. Academic Press, New York, 1967.

[31] S. Saha Ray and P. K. Sahu, Numerical Methods for Solving Fredholm Integral Equations of Second Kind, Abstract and Applied Analysis, Vol. 2013, Article ID 426916, 17 p.

[32] M. J. Emamzadeh and M. T. Kajani, Nonlinear Fredholm Integral Equation of the Second Kind with Quadrature Methods, Journal of E-ISSN: 2224-2880 327 Volume 17, 2018 Mathematical Extension, Vol. 4, No. 2, 2010, pp. 51-58.

[33] L.M.Delves, J.Walsh, Numerical Solution of Integral Equations, Clarendon press, Oxford, 1974.

[34] K.Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM publ., Philadelphia, 1976.

[35] C.T.H.Baker, The Numerical Treatment of Integral Equations, Clarendon Press, Oxford, 1977.

[36] P.J.Davis, P.Rabinowitz, Methods of Numerical Integration. Academic Press, 1984.