A new Bernoulli wavelet method for accurate solutions of nonlinear fuzzy Hammerstein-Volterra delay integral equations

In this article, Bernoulli wavelet method has been developed to solve nonlinear fuzzy Hammerstein-Volterra integral equations with constant delay. This type of integral equation has a particular case the fuzzy variant of a mathematical model from epidemiology. Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The properties of Bernoulli wavelets and Bernoulli polynomials are first presented. The present wavelet method reduces these integral equations to a system of nonlinear algebraic equations and again these algebraic systems have been solved numerically by Newton's method. Convergence analysis of the present method has been discussed in this article. Also the results obtained by present wavelet method have been compared with that of by B-spline wavelet method. Some illustrative examples have been demonstrated to show the applicability and accuracy of the present method.

[1]  H. Sadeghi Goghary,et al.  Two computational methods for solving linear Fredholm fuzzy integral equation of the second kind , 2006, Appl. Math. Comput..

[2]  Alexandru Bica,et al.  A numerical method in terms of the third derivative for a delay integral equation from biomathematics. , 2005 .

[3]  Alexandru Mihai Bica,et al.  Approximating the solution of nonlinear Hammerstein fuzzy integral equations , 2014, Fuzzy Sets Syst..

[4]  Osmo Kaleva.  Fuzzy differential equations , 1987 .

[5]  Lotfi A. Zadeh,et al.  The concept of a linguistic variable and its application to approximate reasoning-III , 1975, Inf. Sci..

[6]  Santanu Saha Ray,et al.  Numerical Methods for Solving Fredholm Integral Equations of Second Kind , 2013 .

[7]  Congxin Wu,et al.  On Henstock integral of fuzzy-number-valued functions (I) , 2001, Fuzzy Sets Syst..

[8]  En-Bing Lin,et al.  Legendre wavelet method for numerical solutions of partial differential equations , 2010 .

[9]  Alexandru Mihai Bica.  One-sided fuzzy numbers and applications to integral equations from epidemiology , 2013, Fuzzy Sets Syst..

[10]  M Barkhordari Ahmadi,et al.  Fuzzy bivariate Chebyshev method for solving fuzzy Volterra-Fredholm integral equations , 2011 .

[11]  Yadollah Ordokhani,et al.  Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations , 2014 .

[12]  Mohammad Ali Fariborzi Araghi,et al.  Numerical solution of fuzzy Fredholm integral equations by the Lagrange interpolation based on the extension principle , 2011, Soft Comput..

[13]  Qingguo Li,et al.  Fuzzy -Continuous Posets , 2013 .

[14]  S. Saha Ray,et al.  Application of Semiorthogonal B-Spline Wavelets for the Solutions of Linear Second Kind Fredholm Integral Equations , 2014 .

[15]  S. Seikkala.  On the fuzzy initial value problem , 1987 .

[16]  Abraham Kandel,et al.  The Cauchy problem for complex fuzzy differential equations , 2014, Fuzzy Sets Syst..

[17]  Hassan Saberi Nik,et al.  Solving linear Fredholm fuzzy integral equations of the second kind by artificial neural networks , 2014 .

[18]  Barnabás Bede,et al.  Quadrature rules for integrals of fuzzy-number-valued functions , 2004, Fuzzy Sets Syst..

[19]  Alexandru Mihai Bica,et al.  Numerical solutions of the nonlinear fuzzy Hammerstein-Volterra delay integral equations , 2013, Inf. Sci..

[20]  Saeid Abbasbandy,et al.  Numerical Solution of Linear Fuzzy Fredholm Integral Equations of the Second Kind Using Fuzzy Haar Wavelet , 2012, IPMU.

[21]  V. Lakshmikantham,et al.  Positive solutions of nonlinear integral equations arising in infectious diseases , 1988 .

[22]  A. Shidfar,et al.  An analytical method for solving linear Fredholm fuzzy integral equations of the second kind , 2011, Comput. Math. Appl..

[23]  Reza Ezzati,et al.  NUMERICAL SOLUTION OF FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND BY USING FUZZY TRANSFORMS , 2012 .

[24]  Shokrollah Ziari,et al.  Numerical Solution And Error Estimation Of Fuzzy Fredholm Integral Equation Using Fuzzy Bernstein Polynomials , 2011 .

[25]  Omran Kouba,et al.  Lecture Notes, Bernoulli Polynomials and Applications , 2013, 1309.7560.

[26]  Hermann Brunner.  Collocation and continuous implicit Runge-Kutta methods for a class of delay Volterra integral equations , 1994 .

[27]  R. Goetschel,et al.  Elementary fuzzy calculus , 1986 .

[28]  Yurilev Chalco-Cano,et al.  Existence of solutions to fuzzy differential equations with generalized Hukuhara derivative via contractive-like mapping principles , 2015, Fuzzy Sets Syst..

[29]  Esmail Babolian,et al.  Numerical method for solving linear Fredholm fuzzy integral equations of the second kind , 2007 .

[30]  Tofigh Allahviranloo,et al.  Numerical Solution of Fuzzy Integral Equations , 2008 .

[31]  Kenneth L. Cooke.  An epidemic equation with immigration , 1976 .

[32]  Azzeddine Bellour,et al.  A Taylor collocation method for solving delay integral equations , 2013, Numerical Algorithms.

[33]  Lotfi A. Zadeh,et al.  On Fuzzy Mapping and Control , 1996, IEEE Trans. Syst. Man Cybern..

[34]  Taher Lotfi,et al.  Fuzzy Galerkin method for solving Fredholm integral equations with error analysis , 2011 .

[35]  Mehdi Dehghan,et al.  Solution of a model describing biological species living together using the variational iteration method , 2008, Math. Comput. Model..

[36]  T Alah Viranlou,et al.  THE HOMOTOPY PERTURBATION METHOD FOR FUZZY FREDHOLM INTEGRAL EQUATIONS , 2008 .

[37]  Santanu Saha Ray,et al.  Numerical solutions for the system of Fredholm integral equations of second kind by a new approach involving semiorthogonal B-spline wavelet collocation method , 2014, Appl. Math. Comput..

[38]  Esmail Babolian,et al.  Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method , 2005, Appl. Math. Comput..

[39]  Sahin Emrah Amrahov,et al.  Solution of linear differential equations with fuzzy boundary values , 2014, Fuzzy Sets Syst..

[40]  Van Hoa Ngo,et al.  Fuzzy fractional functional integral and differential equations , 2015 .

[41]  Peyman Salehi,et al.  NUMERICAL METHOD FOR NON-LINEAR FUZZY VOLTERRA INTEGRAL EQUATIONS OF THE SECOND KIND , 2011 .

[42]  Reza Ezzati,et al.  Iterative method for numerical solution of two-dimensional nonlinear fuzzy integral equations , 2015, Fuzzy Sets Syst..

[43]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[44]  Reza Ezzati,et al.  Numerical solution of nonlinear fuzzy Fredholm integral equations of the second kind using hybrid of block-pulse functions and Taylor series , 2015 .

[45]  Mohammad Keyanpour,et al.  New approach for solving of linear fredholm fuzzy integral equations using sinc function , 2011 .

[46]  Santanu Saha Ray,et al.  Two-dimensional Legendre wavelet method for the numerical solutions of fuzzy integro-differential equations , 2015, J. Intell. Fuzzy Syst..

[47]  Jun Kawabe,et al.  The bounded convergence in measure theorem for nonlinear integral functionals , 2015, Fuzzy Sets Syst..