Hybrid functions approach for the nonlinear Volterra–Fredholm integral equations

Numerical Solution of Nonlinear Volterra-Fredholm Integral Equations Using Haar Wavelet Collocation Method

“…The Haar wavelet function and its operational matrix were employed to solve the resultant integral equations. The numerical results are obtained by the proposed method have been compared with existing methods [7,8]. Illustrative examples are tested with error analysis to justify the efficiency and possibility of the proposed technique.…”

“…The mathematical modelling of many scientific real world problems occurs nonlinearly. Recently, many authors have solved nonlinear VolterraFredholm integral equations from various methods such as Linearization method [7], Hybrid Functions [8], hybrid of block-pulse functions and Taylor series [9], Bessel functions [10], new basis functions [11], radial basis functions [12], Triangular functions (TF) method [13].…”

Solving volterra integral equation via nonlinear programming

“…[1] Integral equation has been one of the principal tools in various areas of applied mathematics, physics and engineering encountered in a variety of applications in many fields including continuum mechanics, potential theory, geophysics, electricity and magnetism, antenna synthesis problem, communication theory, mathematical economics, population genetics and radiation, the particle transport problems of astrophysics and reactor theory, fluid mechanics etc ,In recent years, there has been a growing interest in these mathematical field. [2] Many of integral equations which result from modeling different type of problem are nonlinear , various types of polynomials , have been used by many researchers to develop solutions. Very recently, Maleknejad [3] , Mandal and Bhattacharya [4]and A. Shirin and M. S. Islam [5] used Bernstein polynomials in approximation techniques , Shahsavaran solved by Block Pulse functions [6] Taylor polynomials were also used by Bellour and Rawashdeh [7] and Wang [8] .…”

Solution of the Nonlinear Mixed Volterra-Fredholm Integral Equations by Hybrid of Block-Pulse Functions and Bernoulli Polynomials

“…Several authors consider the nonlinear mixed Volterra-Fredholm integral equations of the form where λ 1 and λ 2 are constants and f ( t ) and the kernels κ 1 ( t , s ) and κ 2 ( t , s ) are given functions assumed to have n th derivatives on the interval 0 ≤ x , t ≤ 1. For the case g 1 ( s , y ( s )) = y p ( s ) and g 2 ( s , y ( s )) = y q ( s ), where p and q are nonnegative integers, Yalçinbaş [ 9 ], Bildik and Inc [ 10 ], and Hashemizadeh et al [ 11 ] used Taylor series, modified decomposition method, and hybrid of block-pulse functions and Legendre polynomials, respectively, to find the solution. For the case g 1 ( s , y ( s )) = F 1 ( y ( s )) and g 2 ( s , y ( s )) = F 2 ( y ( s )), where F 1 ( y ( s )) and F 2 ( y ( s )) are given continuous functions which are nonlinear with respect to y ( s ), Yousefi and Razzaghi [ 12 ] applied Legendre wavelets to obtain the solution, and for the general case, where g 1 ( s , y ( s )) and g 1 ( s , y ( s )) are given continuous functions which are nonlinear with respect to s and y ( s ), Ordokhani [ 13 ] and Marzban et al [ 14 ] applied the rationalized Haar functions and hybrid of block-pulse functions and Lagrange polynomials, respectively, to get the solution.…”

“…Recently, different types of hybrid functions have been used for solving integral equations and proved to be a mathematical power tool [ 8 , 11 , 14 ].…”