Abstract
The principle aim of this paper is to find the numerical solution for system of Volterra–Fredholm integro-differential equations by using the Bernoulli polynomials and the Galerkin method. Through this scheme, the main problem will be transformed to a system of algebraic equations which its solutions are depend on the unknown Bernoulli coefficients. This method gives an analytic solution for systems with polynomial function solution. Better accuracy will be obtained by increasing the number of Bernoulli polynomials. Also, a mathematical proof for its convergence is provided. Moreover, some examples are presented and their numerical results are compared to the results of the Bessel collocation method to show the validity and applicability of this algorithm.
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References
Akyüz A, Sezer M (2003) Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients. Appl Math Comput 144(2–3):237–247
Akyüz-Daşcıoğlu A, Sezer M (2005) Chebyshev polynomial solutions of systems of higher-order linear Fredholm–Volterra integro-differential equations. J Frankl Inst 342(6):688–701
Assari P (2017) The numerical solution of nonlinear integral equations of the second kind using thin plate spline discrete collocation method. Ric Mat 66(2):469–489
Babaaghaie A, Maleknejad K (2017) Numerical solutions of nonlinear two-dimensional partial volterra integro-differential equations by haar wavelet. J Comput Appl Math 317:643–651
Bülbül B, Sezer M (2011) Taylor polynomial solution of hyperbolic type partial differential equations with constant coefficients. Int J Comput Math 88(3):533–544
Chen J, Huang Y, Rong H, Wu T, Zeng T (2015) A multiscale Galerkin method for second-order boundary value problems of fredholm integro-differential equation. J Comput Appl Math 290:633–640
Gülsu M, Gürbüz B, Öztürk Y, Sezer M (2011) Laguerre polynomial approach for solving linear delay difference equations. Appl Math Comput 217(15):6765–6776
Hesameddini E, Asadolhifard E (2013) Solving systems of linear volterra integro-differential equations by using sinc-collocation method. Int J Math Eng Sci 2(7):1–9
Hesameddini E, Rahimi A (2013) A new numerical scheme for solving systems of integro-differential equations. Comput Methods Differ Equ 1(2):108–119
Işik OR, Sezer M, Güney Z (2011) Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel. Appl Math Comput 217(16):7009–7020
Jordán K (1965) Calculus of finite differences, vol 33. American Mathematical Society, Providence
Kheybari S, Darvishi M, Wazwaz AM (2017) A semi-analytical algorithm to solve systems of integro-differential equations under mixed boundary conditions. J Comput Appl Math 317:72–89
Laeli Dastjerdi H, Maalek Ghaini FM (2016) The discrete collocation method for Fredholm–Hammerstein integral equations based on moving least squares method. Int J Comput Math 93(8):1347–1357
Maleknejad K, Kajani MT (2004) Solving linear integro-differential equation system by Galerkin methods with hybrid functions. Appl Math Comput 159(3):603–612
Maleknejad K, Basirat B, Hashemizadeh E (2012) A bernstein operational matrix approach for solving a system of high order linear Volterra–Fredholm integro-differential equations. Math Comput Model 55(3–4):1363–1372
Mokhtary P (2016) Discrete Galerkin method for fractional integro-differential equations. Acta Math Sci 36(2):560–578
Pourabd M et al (2015) Moving least square for systems of integral equations. Appl Math Comput 270:879–889
Pourgholi R, Tahmasebi A, Azimi R (2017) Tau approximate solution of weakly singular Volterra integral equations with Legendre wavelet basis. Int J Comput Math 94(7):1337–1348
Sahu PK, Ray SS (2015a) Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system. Appl Math Comput 256:715–723
Sahu PK, Ray SS (2015b) Numerical solutions for volterra integro-differential forms of Lane-Emden equations of first and second kind using Legendre multi-wavelets. Electron J Diffe Equ 2015(28):1–11
Sekar RCG, Murugesan K (2016) System of linear second order Volterra integro-differential equations using single term Walsh series technique. Appl Mat Computat 273:484–492
Sorkun HH, Yalçinbaş S (2010) Approximate solutions of linear Volterra integral equation systems with variable coefficients. Appl Math Model 34(11):3451–3464
Toutounian F, Tohidi E, Shateyi S (2013) A collocation method based on the bernoulli operational matrix for solving high-order linear complex differential equations in a rectangular domain. In: Abstract and Applied Analysis, vol 2013. Hindawi
Yalçinbaş S, Sezer M, Sorkun HH (2009) Legendre polynomial solutions of high-order linear fredholm integro-differential equations. Appl Math Comput 210(2):334–349
Yunxia W, Yanping C (2017) Legendre spectral collocation method for Volterra–Hammerstein integral equation of the second kind. Acta Math Sci 37(4):1105–1114
Yusufoğlu E (2009) Numerical solving initial value problem for Fredholm type linear integro-differential equation system. J Frankl Inst 346(6):636–649
Yüzbaşı Ş (2015) Numerical solutions of system of linear Fredholm–Volterra integro-differential equations by the Bessel collocation method and error estimation. Appl Math Comput 250:320–338
Yüzbaşı Ş, Sezer M, Kemancı B (2013) Numerical solutions of integro-differential equations and application of a population model with an improved Legendre method. Appl Math Model 37(4):2086–2101
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Hesameddini, E., Riahi, M. Bernoulli Galerkin Matrix Method and Its Convergence Analysis for Solving System of Volterra–Fredholm Integro-Differential Equations. Iran J Sci Technol Trans Sci 43, 1203–1214 (2019). https://doi.org/10.1007/s40995-018-0584-y
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DOI: https://doi.org/10.1007/s40995-018-0584-y
Keywords
- Bernoulli matrix method
- Convergence analysis
- Galerkin method
- Volterra–Fredholm integro-differential systems