A collocation method based on the Bernstein polynomials defined on the interval
The quasilinearization method was introduced by Bellman and Kalaba [
In this paper, we consider the nonlinear FVIDE in the general form
Besides, we approximate to the nonlinear FVIDE (
Bernstein polynomials have many useful properties such as the positivity, continuity, differentiability, integrability, recursion’s relation, symmetry, and unity partition of the basis set over the interval
Now, we give two main theorems for the generalized Berntein polynomials and their basis forms that were proved by Akyuz Dascioglu and Isler [
If
The above theorem can be easily proved by applying transformation
There is a relation between generalized Bernstein basis polynomials matrix and their derivatives in the form
In highlight of these theorems, a collocation method based on the generalized Bernstein polynomials, given in Section
Our aim is to obtain a numerical solution of the nonlinear FVIDE in the general form (
Let
Firstly, by applying the quasilinearization method to the nonlinear FVIDE (
For
For
For
Now we can solve the nonlinear FVDIE (
Firstly, we use Theorem
We need to choose the first iteration function
From expression (
To obtain the solution of nonlinear FVIDE (
If
Four numerical examples are given to illustrate the applicability, accuracy, and efficiency of the proposed method. All results are computed by using an algorithm written in Matlab 7.1. Besides, in the tables, the absolute and
Consider the nonlinear Volterra integrodifferential equation
Let
A numerical comparison of the proposed method with the Hybrid method [
Comparison of the

Presented method  Hybrid method [  
































Consider the nonlinear FredholmVolterra integrodifferential equation
We have two choices satisfying the initial condition for the first iteration functions such that
Table
Comparison of the absolute errors for Example

Presented method  Direct method [ 
Fixed point method [  














































































Consider the thirdorder nonlinear Fredholm integrodifferential equation
Let the first iteration function be
In Table
Absolute errors for Example







































































































































































Consider the fourthorder Volterra integrodifferential equation
Let the first iteration function be
In Table
Comparison of the
Presented method  Hybrid method [  









6 







8 







10 







12 







15 







In general, nonlinear integrodifferential equations can not be solved analytically. For this reason, numerical solutions of nonlinear equations are needed. With the presented method, we have reduced the nonlinear FVIDE (
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by Scientific Research Project Coordination Unit of Pamukkale University with no. 2012FBE036.