[Retracted] Numerical Solution of Nonlinear Fredholm Integrodifferential Equations by Hybrid of Block-Pulse Functions and Normalized Bernstein Polynomials

Abstract

A numerical method for solving nonlinear Fredholm integrodifferential equations is proposed. The method is based on hybrid functions approximate. The properties of hybrid of block pulse functions and orthonormal Bernstein polynomials are presented and utilized to reduce the problem to the solution of nonlinear algebraic equations. Numerical examples are introduced to illustrate the effectiveness and simplicity of the present method.

1. Introduction

Integrodifferential equations are often involved in mathematical formulation of physical phenomena. Fredholm integrodifferential equations play an important role in many fields such as economics, biomechanics, control, elasticity, fluid dynamics, heat and mass transfer, oscillation theory, and airfoil theory; for examples see [1–3] and references cited therein. Finding numerical solutions for Fredholm integrodifferential equations is one of the oldest problems in applied mathematics. Numerous works have been focusing on the development of more advanced and efficient methods for solving integrodifferential equations such as wavelets method [4, 5], Walsh functions method [6], sinc-collocation method [7], homotopy analysis method [8], differential transform method [9], the hybrid Legendre polynomials and block-pulse functions [10], Chebyshev polynomials method [11], and Bernoulli matrix method [12].

Block-pulse functions have been studied and applied extensively as a basic set of functions for signals and functions approximations. All these studies and applications show that block-pulse functions have definite advantages for solving problems involving integrals and derivatives due to their clearness in expressions and their simplicity in formulations; see [13]. Also, Bernstein polynomials play a prominent role in various areas of mathematics. Many authors have used these polynomials in the solution of integral equations, differential equations, and approximation theory; see for instance [14–17].

The purpose of this work is to utilize the hybrid functions consisting of combination of block-pulse functions with normalized Bernstein polynomials for obtaining numerical solution of nonlinear Fredholm integrodifferential equation: ^𝑠_𝑖=0𝑝_𝑖(𝑥)𝑦^(𝑖)(𝑥)=𝑔(𝑥)+𝜆¹₀𝑘(𝑥,𝑡)𝑦(𝑡)^𝑞𝑑𝑡,0≤𝑥,𝑡<1, (1) with the conditions 𝑦^(𝑖)(0)=𝛼_𝑖,0≤𝑖≤𝑠−1, (2) where 𝑦^(𝑖)(𝑥) is the 𝑖th derivative of the unknown function that will be determined, 𝑘(𝑥,𝑡) is the kernel of the integral, 𝑔(𝑥) and 𝑝_𝑖(𝑥) are known analytic functions, 𝑞 is a positive integer, and 𝜆 and 𝛼_𝑖 are suitable constants. The proposed approach for solving this problem uses few numbers of basis and benefits of the orthogonality of block-pulse functions and the advantages of orthonormal Bernstein polynomials properties to reduce the nonlinear integrodifferential equation to easily solvable nonlinear algebraic equations.

This paper is organized as follows. In the next section, we present Bernstein polynomials and hybrid of block-pulse functions. Also, their useful properties such as functions approximation, convergence analysis, operational matrix of product, and operational matrix of differentiation are given. In Section 3, the numerical scheme for the solution of (1) and (2) is described. In Section 4, the proposed method is applied to some nonlinear Fredholm integrodifferential equations, and comparisons are mad with the existing analytic or numerical solutions that were reported in other published works in the literature. Finally conclusions are given in Section 5.

2. Properties of Hybrid Functions

2.1. Hybrid of Block-Pulse Functions and Orthonormal Bernstein Polynomials

The Bernstein polynomials of 𝑛th degree are defined on the interval [0,1] as [16] 𝐵_𝑖,𝑛(𝑥)=𝑛𝑖𝑥^𝑖(1−𝑥)^𝑛−𝑖,for𝑖=0,1,2,…,𝑛, (3) where 𝑛𝑖=𝑛!()!.𝑖!𝑛−𝑖 (4) There are (𝑛+1)  𝑛th degree Bernstein polynomials. Using Gram-Schmidt orthonormalization process on 𝐵_𝑖,𝑛(𝑥), we obtain a class of orthonormal polynomials from the Bernstein polynomials. We call them orthonormal Bernstein polynomials of degree 𝑛 and denote them by 𝑏_𝑖,𝑛(𝑥), 0≤𝑖≤𝑛. For 𝑛=3, the four orthonormal Bernstein polynomials are given by 𝑏_0,3(𝑥)=−√7𝑥³−3𝑥²,𝑏+3𝑥−1_1,3(𝑥)=√57𝑥³−15𝑥²,𝑏+9𝑥−1_2,3(𝑥)=−√321𝑥³−33𝑥²,𝑏+13𝑥−1_3,3(𝑥)=35𝑥³−45𝑥²+15𝑥−1. (5)

Hybrid functions ℎ_𝑗𝑖(𝑥),𝑗=1,2,…,𝑚 and 𝑖=0,1,…,𝑛 are defined on the interval [0,1) as ℎ_𝑗𝑖(𝑥)=√𝑚𝑏_𝑖,𝑛𝑚𝑥−𝑗+1,𝑗−1𝑚𝑗≤𝑥<𝑚,0,otherwise, (6) where 𝑗 and 𝑛 are the order of block-pulse functions and degree of orthonormal Bernstein polynomials, respectively.

It is clear that these sets of hybrid functions in (6) are orthonormal and disjoint.

2.2. Functions Approximation

A function 𝑦(𝑥)∈𝐿²[0,1) may be approximated as 𝑦(𝑥)≈^𝑚𝑛_𝑗=1𝑖=0𝑐_𝑗𝑖ℎ_𝑗𝑖(𝑥)=𝐂^𝑇𝐇(𝑥), (7) where 𝐂=𝐂^𝑇₁,𝐂^𝑇₂,…,𝐂^𝑇_𝑗,…,𝐂^𝑇_𝑚^𝑇,𝐂_𝑗=𝑐_𝑗0,𝑐_𝑗1,𝑐_𝑗2,…,𝑐_𝑗𝑛^𝑇,𝑗=1,2,…,𝑚,𝐇(𝑥)=𝐇^𝑇₁(𝑥),𝐇^𝑇₂(𝑥),…,𝐇^𝑇_𝑗(𝑥),…,𝐇^𝑇_𝑚(𝑥)^𝑇, (8)(9) and 𝐇_𝑗(𝑥)=[ℎ_𝑗0(𝑥),ℎ_𝑗1(𝑥),…,ℎ_𝑗𝑛(𝑥)]^𝑇, 𝑗=1,2,…,𝑚. The constant coefficients 𝑐_𝑗𝑖 are (𝑦(𝑥),ℎ_𝑗𝑖(𝑥)),  𝑖=0,1,2,…,𝑛, 𝑗=1,2,…,𝑚, and (⋅,⋅) is the standard inner product on 𝐿²[0,1).

We can also approximate the function 𝑘(𝑥,𝑡)∈𝐿²([0,1)×[0,1)) by 𝑘(𝑥,𝑡)≈^𝑚𝑚_𝑖=1^𝑛_𝑗=1^𝑛_𝑙=0𝑟=0𝑘^𝑖𝑗_𝑙𝑟ℎ_𝑖𝑙(𝑥)ℎ_𝑗𝑟(𝑡)=𝐇^𝑇(𝑥)𝐊𝐇(𝑡), (10) where 𝐊=[𝐊^𝑖𝑗] is an 𝑚(𝑛+1)×𝑚(𝑛+1) matrix, such that the elements of the sub matrix 𝐤^𝑖𝑗 are 𝑘^𝑖𝑗_𝑙𝑟=^𝑖/𝑚_{𝑖−1/𝑚}^𝑗/𝑚_{𝑗−1/𝑚}𝑘(𝑥,𝑡)ℎ_{𝑖(𝑙−1)}(𝑥)ℎ_{𝑗(𝑟−1)}(𝑡)𝑑𝑥𝑑𝑡,𝑙,𝑟=1,2,…,𝑛+1,𝑖,𝑗=1,2,…,𝑚, (11) utilizing properties of block-pulse function and orthonormal Bernstein polynomials.

2.3. Convergence Analysis

In this section, the error bound and convergence is established by the following lemma.

Lemma 1. Suppose that 𝑓∈𝐶^(𝑛+1)[0,1) is 𝑛+1 times continuously differentiable function such that 𝑓=∑^𝑚_𝑗=1𝑓_𝑗, and let 𝑌_𝑗=Span{ℎ_𝑗0(𝑥),ℎ_𝑗1(𝑥),…,ℎ_𝑗𝑛(𝑥)}, 𝑗=1,2,…,𝑚. If 𝐂^𝑇_𝑗𝐇_𝑗(𝑥) is the best approximation to 𝑓_𝑗 from 𝑌_𝑗, then 𝐂^𝑇𝐇(𝑥) approximates 𝑓 with the following error bound: 𝑓−𝐂^𝑇𝐇(𝑥)₂≤𝛾𝑚^𝑛+1(𝑛+1)!√,2𝑛+3𝛾=max_[)𝑥∈0.1𝑓^(𝑛+1)(𝑥). (12)

Proof. The Taylor expansion for the function 𝑓_𝑗(𝑥) is 𝑓_𝑗(𝑥)=𝑓_𝑗𝑗−1𝑚+𝑓^′_𝑗𝑗−1𝑚𝑥−𝑗−1𝑚+⋯+𝑓_𝑗^(𝑛)𝑗−1𝑚𝑥−𝑗−1/𝑚^𝑛,𝑛!𝑗−1𝑚𝑗≤𝑥<𝑚, (13) for which it is known that 𝑓_𝑗(𝑥)−𝑓_𝑗(𝑥)≤𝑓^(𝑛+1)𝜂𝑥−𝑗−1/𝑚^𝑛+1()!,𝑛+1𝜂∈𝑗−1𝑚,𝑗𝑚,𝑗=1,2,…,𝑚. (14) Since 𝐂^𝑇_𝑗𝐇_𝑗(𝑥) is the best approximation to 𝑓_𝑗 form 𝑌_𝑗 and 𝑓_𝑗∈𝑌_𝑗, using (14) we have 𝑓_𝑗−𝐂^𝑇_𝑗𝐇_𝑗(𝑥)²₂≤𝑓_𝑗−𝑓_𝑗²=^𝑗/𝑚_{𝑗−1/𝑚}𝑓_𝑗(𝑥)−𝑓_𝑗(𝑥)²≤𝑑𝑥_{𝑗/𝑚𝑗−1/𝑚}𝑓^(𝑛+1)𝜂𝑥−𝑗−1/𝑚^𝑛+1(𝑛+1)!²≤𝛾𝑑𝑥(𝑛+1)!^2𝑗/𝑚_{𝑗−1/𝑚}𝑥−𝑗−1𝑚^2𝑛+2=𝛾𝑑𝑥(𝑛+1)!²1𝑚^2𝑛+3().2𝑛+3 (15) Now, 𝑓−𝐂^𝑇𝐇(𝑥)²₂≤^𝑚_𝑗=1𝑓_𝑗−𝐂^𝑇_𝑗𝐇_𝑗(𝑥)²₂≤𝛾²𝑚^2𝑛+2[(𝑛+1)!]²().2𝑛+3 (16) By taking the square roots we have the above bound.

2.4. The Operational Matrix of Product

In this section, we present a general formula for finding the 𝑚(𝑛+1)×𝑚(𝑛+1) operational matrix of product 𝐂 whenever 𝐂^𝑇𝐇(𝑥)𝐇^𝑇(𝑥)≈𝐇^𝑇(𝑥)𝐂, (17) where 𝐂=diag𝐂₁,𝐂₂,…,𝐂_𝑗,…,𝐂_𝑚. (18) In (18), 𝐂_𝑗=[𝑐^𝑗_𝑙𝑟] are (𝑛+1)×(𝑛+1) symmetric matrices depending on 𝑛, where 𝑐^𝑗_𝑙𝑟=^𝑗/𝑚_{𝑗−1/𝑚}ℎ_{𝑗(𝑙−1)}(𝑥)ℎ_{𝑗(𝑟−1)}(𝑥)^𝑛_𝑖=0𝑐_𝑗𝑖ℎ_𝑗𝑖(𝑥)𝑑𝑥,𝑙,𝑟=1,2,…,𝑛+1. (19)

Furthermore, the integration of cross-product of two hybrid functions vectors is ¹₀𝐇(𝑥)𝐇^𝑇(𝑥)𝑑𝑥=𝐈, (20) where 𝐈 is the 𝑚(𝑛+1) identity matrix.

2.5. The Operational Matrix of Differentiation

The operational matrix of derivative of the hybrid functions vector 𝐇(𝑥) is defined by 𝑑𝐇(𝑥)(𝑥),𝑑𝑥=𝐃𝐇 (21) where 𝐃 is the 𝑚(𝑛+1)×𝑚(𝑛+1) operational matrix of derivative given as 𝐇(𝑥)=𝐇^𝑇₁(𝑥),𝐇^𝑇₂(𝑥),…,𝐇^𝑇_𝑗(𝑥),…,𝐇^𝑇_𝑚(𝑥)^𝑇=𝐀𝐓(𝑥), (22) where 𝐀=diag[𝐀₁,𝐀₂,…,𝐀_𝑗,…,𝐀_𝑚] is the 𝑚(𝑛+1)×𝑚(𝑛+1) coefficient matrix of the (𝑛+1)×(𝑛+1) coefficient submatrix 𝐀_𝑗, and 𝐓(𝑥)=[𝐭₁(𝑥),𝐭₂(𝑥),…,𝐭_𝑗(𝑥),…,𝐭_𝑚(𝑥)]^𝑇 is the 𝑚(𝑛+1) vector with 𝐭_𝑗(𝑥)=[1,𝑥,𝑥²,…,𝑥^𝑛]^𝑇, such that 𝐇_𝑗(𝑥)=𝐀_𝑗𝐭_𝑗(𝑥). Now 𝑑𝐇(𝑥)=𝐀𝐐𝐓(𝑥)=𝐀𝐐𝐀𝑑𝑥⁻¹𝐇(𝑥), (23) where 𝐐=diag[𝐐,…,𝐐] is the 𝑚(𝑛+1)×𝑚(𝑛+1) matrix of the (𝑛+1)×(𝑛+1) sub-matrix 𝐐, such that 𝐐=.000⋯00100⋯00020⋯00⋮⋮⋮⋯⋮⋮000⋯𝑛0 (24) Hence, 𝐃=𝐀𝐐𝐀⁻¹. (25) In general, we can have 𝑑^𝑘𝑑𝑥^𝑘𝐇(𝑥)=𝐃^𝑘𝐇(𝑥),𝑘=1,2,3,…. (26)

3. Outline of the Solution Method

This section presents the derivation of the method for solving 𝑠th-order nonlinear Fredholm integrodifferential equation (1) with the initial conditions (2).

Step  1. The functions 𝑦^(𝑖)(𝑥), 𝑖=0,1,2,…,𝑠 are being approximated by 𝑦^(𝑖)(𝑥)=𝐂^𝑇(𝐇(𝑥))^(𝑖)=𝐂^𝑇𝐃^𝑖𝐇(𝑥),𝑖=0,1,2,…,𝑠, (27) where 𝐃 is given by (25).

Step  2. The function 𝑘(𝑥,𝑡) is being approximated by (10).

Step  3. In this step, we present a general formula for approximate 𝑦^𝑞(𝑥). By using (7) and (17), we can have 𝑦²(𝑥)=𝐂^𝑇𝐇(𝑥)²=𝐂^𝑇𝐇(𝑥)𝐇^𝑇(𝑥)𝐂=𝐇^𝑇(𝑥)𝐂𝐂,𝑦³(𝑥)=𝐂^𝑇𝐇(𝑥)𝐂^𝑇𝐇(𝑥)²=𝐂^𝑇𝐇(𝑥)𝐇^𝑇(𝑥)𝐂𝐂=𝐇^𝑇(𝑥)𝐂𝐂𝐂=𝐇^𝑇(𝑥)𝐂²𝐂, (28)(29) and so by use of induction, 𝑦^𝑞(𝑥) will be approximated as 𝑦^𝑞(𝑥)=𝐇^𝑇(𝑥)𝐂^𝑞−1𝐂. (30)

Step  4. Approximate the functions 𝑔(𝑥) and 𝑝_𝑖(𝑥) by 𝑔(𝑥)≈𝐆^𝑇𝐇(𝑥),𝑝_𝑖(𝑥)≈𝐏^𝑇_𝑖𝐻(𝑥),𝑖=0,1,2,…,𝑠, (31)(32) where 𝐆 and 𝐏_𝑖 are constant coefficient vectors which are defined similarly to (7).

Now, using (27)–(32) and (10) to substitute into (1), we can obtain ^𝑠_𝑖=0𝐏^𝑇_𝑖𝐇(𝑥)𝐇^𝑇(𝑥)𝐃^𝑖𝑇𝐂=𝐇^𝑇(𝑥)𝐆+𝜆¹₀𝐇^𝑇(𝑥)𝐊𝐇(𝑡)𝐇^𝑇(𝑡)𝐂^𝑞−1𝐂𝑑𝑡. (33) Utilizing (17) and (20), we may have ^𝑠_𝑖=0𝐇^𝑇(𝑥)𝐏_𝑖𝐃^𝑖𝑇𝐂=𝐇^𝑇(𝑥)𝐆+𝜆𝐇^𝑇(𝑥)𝐊𝐂^𝑞−1𝐂, (34) and hence we get ^𝑠_𝑖=0𝐏_𝑖𝐃^𝑖𝑇𝐂−𝜆𝐊𝐂^𝑞−1𝐂=𝐆. (35) The matrix (35) gives a system of 𝑚(𝑛+1) nonlinear algebraic equations which can be solved utilizing the initial condition for the elements of 𝐂. Once 𝐂 is known, 𝑦(𝑥) can be constructed by using (7).

4. Applications and Numerical Results

In this section, numerical results of some examples are presented to validate accuracy, applicability, and convergence of the proposed method. Absolute difference errors of this method is compared with the existing methods reported in the literature [5, 6, 17, 18]. The computations associated with these examples were performed using MATLAB 9.0.

Example 1. Consider the first-order nonlinear Fredholm integrodifferential equation [17, 18] as follows: 𝑦^′(𝑥)1=1−3𝑥+¹₀𝑥𝑦²(𝑡)𝑑𝑡,0≤𝑥<1, (36) with the initial condition 𝑦(0)=0. (37) In this example, we have 𝑝₀=0, 𝑝₁=1, 𝑔(𝑥)=1−(1/3)𝑥, 𝜆=1, 𝑘(𝑥,𝑡)=𝑥, and 𝑞=2.
The matrix (35) for this example is 𝐏₁𝐃^𝑇𝐂−𝐊𝐂𝐂=𝐆, (38) where for 𝑛=1 and 𝑚=2 we have𝐏₁=𝐈,𝐃^𝑇=−33√−√300√330000−33300−√𝑐33,𝐂=¹⁰𝑐₁₁𝑐₂₀𝑐₂₁,1𝐊=√163148√163√483116√1631161164√31124√3√123818√3818,1𝐂=43√6𝑐₁₀−√2𝑐₁₁−√2𝑐₁₀+√6𝑐₁₁−√002𝑐₁₀+√6𝑐₁₁√6𝑐₁₀+5√2𝑐₁₁√000036𝑐₂₀−√2𝑐₂₁−√2𝑐₂₀+√6𝑐₂₁00−√2𝑐₂₀+√6𝑐₂₁√6𝑐₂₀+5√2𝑐₂₁,√𝐆=1765√7227√246√3626. (39)Equation (38) gives a system of nonlinear algebraic equations that can be solved utilizing the initial condition (37); that is, √6𝑐₁₀−√2𝑐₁₁=0, we obtain 𝑐₁₀=√624,𝑐₁₁=√28,𝑐₂₀=√66,𝑐₂₁=√24. (40) Substituting these values into (7), the result will be 𝑦(𝑥)=𝑥, that is, the exact solution. It is noted that the result gives the exact solution as in [17], while in [18] using the sinc method the maximum absolute error is 1.52165×10⁻³.

Example 2. Consider the first-order nonlinear Fredholm integrodifferential equation [6, 17] as follows: 𝑥𝑦^′(𝑥)(𝑥)1−𝑦=−6+45𝑥²+¹₀𝑥²+𝑡𝑦²(𝑡)𝑑𝑡,0≤𝑥<1. (41) with the initial condition 𝑦(0)=0. (42) In this example, we have 𝑝₀=−1, 𝑝₁=𝑥, 𝑔(𝑥)=−(1/6)+(4/5)𝑥², 𝜆=1, 𝑘(𝑥,𝑡)=𝑥²+𝑡, and 𝑞=2.
The matrix (35) for this example is 𝐏₀+𝐏₁𝐃^𝑇𝐂−𝐊𝐂𝐂=𝐆, (43) where for 𝑛=2 and 𝑚=2 we have𝐏₀=−𝐈,𝐏₁=1√1215−√605√120000151604√3−√240005√1203524712000000√1215−√605√120000153604√3−√240005√12032411,𝐃12_𝑇=7√−5153−2√−√5000153√−314330√000−8337√8000000−5153−2√5−√000153√−31433√0000−8338𝑐,𝐂=¹⁰𝑐₁₁𝑐₁₂𝑐₂₀𝑐₂₁𝑐₂₂−√,𝐆=1110−√4506√902√1802310√900136√180192,1180𝐊=√24157√45524013√7215√20415√720151725√123√14415124√163√1655√48311447√245√1443516772√483115√720152041√14417157√24057√901531445√163√721115144137√483√725√161131144√121355√14431489,𝐜̃𝐂=¹𝟎𝟎𝐜̃_2̃𝐜𝑗,=^{5√107𝑐𝑗0−5√6𝑐21𝑗1+√27𝑐𝑗2−5√6𝑐21𝑗0+√1110𝑐35𝑗1−8√30𝑐105𝑗2√27𝑐𝑗0−8√30𝑐105𝑗1+3√10𝑐35𝑗2}₋^5√6_𝑐21𝑗0+^√1110_{𝑐35𝑗1−}^8√30_𝑐105𝑗2^√1110_𝑐35𝑗0+^3√6_7𝑐𝑗1+^√2_{7𝑐𝑗2−}^8√30_{𝑐105𝑗0+}^√2_7𝑐𝑗1+^5√6_{𝑐21𝑗2√27𝑐𝑗0−8√30𝑐105𝑗1+3√10𝑐35𝑗2−8√30𝑐105𝑗0+√27𝑐𝑗1+5√6𝑐21𝑗23√10𝑐35𝑗0+5√6𝑐21𝑗1+√1327𝑐𝑗2},𝑗=1,2. (44)

Equation (43) gives a system of nonlinear algebraic equations that can be solved utilizing the initial condition (42); that is, √10𝑐₁₀−√6𝑐₁₁+√2𝑐₁₂=0, we obtain 𝑐₁₀=√10240,𝑐₁₁=√6,𝑐48₁₂=√224,𝑐₂₀=√10,𝑐15₂₁=√68,𝑐₂₂=√26. (45) Substituting these values into (7), the result will be 𝑦(𝑥)=𝑥², that is, the exact solution. It is noted that the result gives the exact solution as in [17], while in [6] approximate solution is obtained with maximum absolute error 1.0000×10⁻⁵.

Example 3. Consider the second-order nonlinear Fredholm integrodifferential equation [17] as follows: 𝑦^′′(𝑥)+𝑥𝑦^′(𝑥)−𝑥𝑦(𝑥)=𝑒^𝑥sin𝑥+¹₀sin𝑥⋅𝑒^−2𝑡𝑦²(𝑡)𝑑𝑡,0≤𝑥<1, (46) with the initial conditions 𝑦(0)=𝑦^′(0)=1. (47) The exact solution is 𝑦(𝑥)=𝑒^𝑥. We solve this example by using the proposed method with 𝑛=2, 𝑚=30 and  𝑛=3, 𝑚=30. Comparison among the proposed method and methods in [17] is shown in Table 1. It is clear from this table that the results obtained by the proposed method, using few numbers of basis, are very promising and superior to that of [17].

Table 1 

Numerical comparison of absolute difference errors for Example 3.

Example 4. Consider the following nonlinear Fredholm integrodifferential equation [5, 17]: 𝑦^′(𝑥)(𝑥)=1+𝑦2𝑒⁻²−1+¹₀𝑦²(𝑡)𝑑𝑡,0≤𝑥<1, (48) with the initial conditions 𝑦(0)=1. (49) The exact solution of this problem is 𝑦(𝑥)=𝑒^−𝑥. In Table 2 we have compared the absolute difference errors of the proposed method with the collocation method based on Haar wavelets in [5] and method in [17].
Maximum absolute errors of Example 4 for some different values of 𝑛 and 𝑚 are shown in Table 3. As it is seen from Table 3, for a certain value of 𝑛 as 𝑚 increases the accuracy increases, and for a certain value of 𝑚 as 𝑛 increases the accuracy increases as well. In case of 𝑚=1, the numerical solution obtained is based on orthonormal Bernstein polynomials only, while in case of 𝑛=0, the numerical solution obtained is based on block-pulse functions only.

Table 2 

Numerical comparison of absolute difference errors for Example 4.

Table 3 

Maximum absolute errors for different values of 𝑛 and 𝑚 for Example 4.

5. Conclusion

In this work, we present a numerical method for solving nonlinear Fredholm integrodifferential equations based on hybrid of block-pulse functions and normalized Bernstein polynomials. One of the most important properties of this method is obtaining the analytical solutions if the equation has an exact solution, that is, a polynomial function. Another considerable advantage is this method has high relative accuracy for small numbers of basis 𝑛. The matrices 𝐊, 𝐂, and 𝐃 in (10), (17), and (25), respectively, have large numbers of zero elements, and they are sparse; hence, the present method is very attractive and reduces the CPU time and computer memory. Moreover, satisfactory results of illustrative examples with respect to several other methods (e.g., Haar wavelets method, Walsh functions method, Bernstein polynomials method, and sinc collocation method) are included to demonstrate the validity and applicability of the proposed method.