Fibonacci Collocation Method for Solving High-Order Linear Fredholm Integro-Differential-Difference Equations

Abstract

A new collocation method based on the Fibonacci polynomials is introduced for the approximate solution of high order-linear Fredholm integro-differential-difference equations with the mixed conditions. The proposed method is analyzed to show the convergence of the method. Some further numerical experiments are carried out to demonstrate the method.

1. Introduction

The integro-differential-difference equations (IDDEs) have been developed very rapidly in recent years. This is an important branch of mathematics which has a lot of interest in many application fields such as engineering, mechanics, physics, astronomy, chemistry, biology, economics, and potential theory, electrostatics [1–14]. Since some IDDEs are hard to solve numerically, they are solved by using the approximated methods. Several numerical methods were used such as the successive approximations, Adomian decomposition, Haar Wavelet, and Tau and Walsh series methods [15–20]. Additionally the Monte Carlo method for linear Fredholm integro-differential-difference equation has been presented by Farnoosh and Ebrahimi [21] and the Direct method based on the Fourier and block-pulse method functions by Asady et al. [22].

Since the beginning of 1994, the Taylor and Chebyshev matrix methods have also been used by Sezer et al. to solve linear differential, Fredholm integral, and Fredholm integro-differential equations [23–35]. Lately, the Fibonacci collocation method has been used to find the approximate solutions of differential, integral, and integro-differential equations [36].

In this study, we consider the approximate solution of the 𝑚th-order Fredholm integro-differential-difference equations, ^𝑚_𝑘=0𝑃_𝑘(𝑥)𝑦^(𝑘)(𝑥)+^𝑠_𝑟=0𝑄_𝑟(𝑥)𝑦^(𝑟)𝜇_𝑟𝑥+𝜏_𝑟=g(𝑥)+𝜆^𝑏_𝑎𝐾(𝑥,𝑡)𝑦(𝑡)𝑑𝑡, (1) where 𝑠≤𝑚, 𝜏_𝑟 are the integer,0≤𝑎≤𝑥, 𝑡≤𝑏, under the mixed conditions ^𝑚−1_𝑘=0𝑎_𝑗𝑘𝑦^(𝑘)(𝑎)+𝑏_𝑗𝑘𝑦^(𝑘)(𝑏)=𝜆_𝑗,𝑗=1,2,3,…,𝑚, (2) where 𝑃_𝑘(𝑥), 𝑄_𝑟(𝑥),𝑔(𝑥), and 𝐾(𝑥,𝑡)are functions defined on 𝑎≤𝑥, 𝑡≤𝑏; 𝑎_𝑗𝑘, 𝑏_𝑗𝑘, 𝜆, and 𝜆_𝑗 are suitable constants.

Our aim is to obtain an approximate solution of (1) expressed in the truncated Fibonacci series form: 𝑦(𝑥)=^𝑁_𝑛=1𝑎_𝑛𝐹_𝑛(𝑥), (3) where 𝑎_𝑛, 𝑛=1,2,3,…,𝑁, are the unknown Fibonacci coefficients. Here 𝑁 is positive integer such that 𝑁≥𝑚 and 𝐹_𝑛(𝑥), 𝑛=1,2,3,…,𝑁, are the Fibonacci polynomials defined by 𝐹_𝑛(𝑥)=^{[(𝑛−1)/2]}_𝑗=0𝑗𝑥𝑛−𝑗−1_{𝑛−2𝑗−1},(𝑛−1)2=(𝑛−2)2(),𝑛even,𝑛−12,𝑛odd. (4)

2. Fundamental Matrix Relations

Firstly, we can write the Fibonacci polynomials 𝐹_𝑛(𝑥) in the matrix form as follows: 𝐅^𝑇(𝑥)=𝐂𝐗^𝑇(𝑥)⟺𝐅(𝑥)=𝐗(𝑥)𝐂^𝑇, (5) where 𝐅(𝑥)=𝐹₁(𝑥)𝐹₂(𝑥)⋯𝐹_𝑁(𝑥),𝐗(𝑥)=1𝑥⋯𝑥^𝑁−1. (6) If 𝑁 is even, 𝐂=^0001011020₀²₁₀³₀₍₎^{000⋯000⋯00⋯0}_{⋯0⋮⋮⋮⋮⋮⋮𝑛−22()𝑛−220𝑛2()𝑛−420𝑛0⋯02()𝑛−220()𝑛+22()𝑛−42⋯0𝑛−1𝑁×𝑁}; (7)if 𝑁 is odd, 𝐂=^0001011020₀²₁₀³₀₀₍₎^{000⋯000⋯00⋯0}_{⋯0⋮⋮⋮⋮⋮⋮𝑛−12()𝑛−320()𝑛+12()𝑛−52()⋯0𝑛−12()𝑛−120()𝑛+12()𝑛−320⋯⋯𝑛−1𝑁×𝑁}. (8)

Let us show (1) in the following form: 𝑃(𝑥)+𝑄(𝑥)=g(𝑥)+𝜆𝐼(𝑥), (9) where 𝑃(𝑥)=^𝑚_𝑘=0𝑃_𝑘(𝑥)𝑦^(𝑘)(𝑥),𝑄(𝑥)=^𝑠_𝑟=0𝑄_𝑟(𝑥)𝑦^(𝑟)𝜇_𝑟𝑥+𝜏_𝑟,𝐼(𝑥)=𝜆^𝑏_𝑎𝐾(𝑥,𝑡)𝑦(𝑡)𝑑𝑡. (10)

2.1. Matrix Relations for the Differential Part 𝑃(𝑥)

Firstly, we consider the solution 𝐲(𝑥) and its 𝑘th derivate 𝐲^(𝑘)(𝑥) in the matrix form: 𝐲(𝑥)=𝐅(𝑥)𝐀,𝐀=𝑎₁𝑎₂⋯𝑎_𝑁^𝑇,𝐲^(𝑘)(𝑥)=𝐅^(𝑘)(𝑥)𝐀. (11)(12) Then, from relations (5) and (11), we can obtain the following matrix form: 𝐲(𝑥)=𝐗(𝑥)𝐂^𝑇𝐀. (13) Similar to (13), from relations (5), (11), and (12), we can find 𝐲^(𝑘)(𝑥)matrix form as 𝐲^(𝑘)(𝑥)=𝐗^(𝑘)(𝑥)𝐂^𝑇𝐀. (14) To find the matrix 𝐗^(𝑘)(𝑥)in terms of the matrix𝐗(𝑥), we can use the following relation: 𝐗⁽¹⁾(𝑥)=𝐗(𝑥)𝐓^𝑇,𝐗⁽²⁾(𝑥)=𝐗⁽¹⁾(𝑥)𝐓^𝑇=𝐗(𝑥)𝐓^𝑇𝐓^𝑇=𝐗(𝑥)𝐓^𝑇2⋮𝐗^(𝑘)(𝑥)=𝐗^(𝑘−1)(𝑥)𝐓^𝑇=𝐗(𝑥)𝐓^𝑇𝑘, (15) where 𝐓^𝑇=010⋯000002⋯000000⋯000⋮⋮⋮⋮⋮000⋯00𝑁−1000⋯000. (16) Subsequently, by substituting the matrix form (15) into (14), we obtain the matrix relations 𝐲^(𝑘)(𝑥)=𝐗(𝑥)𝐓^𝑇𝑘𝐂^𝑇𝐀. (17)

2.2. Matrix Relations for the Difference Part 𝑄(𝑥)

If we put 𝑥→𝜇_𝑟𝑥+𝜏_𝑟 in the relation (11), we have the matrix form 𝐲𝜇_𝑟+𝜏_𝑟=𝐅𝜇_𝑟𝑥+𝜏_𝑟𝐀. (18) It is seen that the relation between the matrices𝐗(𝑥)and𝐗(𝜇_𝑟𝑥+𝜏_𝑟)is 𝐗𝜇_𝑟𝑥+𝜏_𝑟=𝐗(𝑥)𝜷𝜇_𝑟,𝜏_𝑟, (19) where 𝜷𝜇_𝑟,𝜏_𝑟=00𝜇_𝑟⁰𝜏_𝑟⁰10𝜇_𝑟⁰𝜏_𝑟¹⋯0𝜇𝑁−1_𝑟0𝜏_𝑟^𝑁−1011𝜇_𝑟¹𝜏_𝑟⁰⋯1𝜇𝑁−1_𝑟1𝜏_𝑟^𝑁−2𝜇⋮⋮⋱⋮00⋯𝑁−1𝑁−1^𝑟𝑁−1𝜏_𝑟⁰. (20) By using the relations (15) and (19), we can get 𝐗^(𝑘)𝜇_𝑟𝑥+𝜏_𝑟=𝐗(𝑥)𝜷𝜇_𝑟,𝜏_𝑟𝐓^𝑇𝑘. (21) Thus from (14) and (21), we can find 𝐲^(𝑘)𝜇_𝑟+𝜏_𝑟=𝐗(𝑥)𝜷𝜇_𝑟,𝜏_𝑟𝐓^𝑇𝑘𝐂^𝑇𝐀. (22) By using the expressions (14) and (22), we obtain the matrix form 𝑃(𝑥)=^𝑚_𝑘=0𝑝_𝑘(𝑥)𝐗(𝑥)𝐓^𝑇𝑘𝐂^𝑇𝑄(𝑥)=𝐀,_𝑠𝑟=0𝑄_𝑟(𝑥)𝐗(𝑥)𝜷𝜇_𝑟,𝜏_𝑟𝐓^𝑇𝑟𝐂^𝑇𝐀. (23)

2.3. Matrix Relations for the Integral Part

Let us find the matrix relation for the Fredholm integral part 𝐼(𝑥)in (9). The kernel function 𝐾(𝑥,𝑡)can be shown by the truncated Fibonacci series, 𝐾(𝑥,𝑡)=^𝑁𝑁_𝑚=0𝑛=0𝑘^𝑓_𝑚𝑛𝐹_𝑚(𝑥)𝐹_𝑛(𝑡), (24) and the truncated Taylor series, 𝐾(𝑥,𝑡)=^𝑁𝑁_𝑚=0𝑛=0𝑘^𝑡_𝑚𝑛𝑥^𝑚𝑡^𝑛, (25) where 𝑘^𝑡_𝑚𝑛=1𝜕𝑚!𝑛!^𝑚+𝑛𝐾(0,0)𝜕𝑥^𝑚𝜕𝑥^𝑛;𝑚,𝑛=0,1,…,𝑁. (26) The expressions (24) and (25) can be put in matrix forms as 𝐾(𝑥,𝑡)=𝐅(𝑥)𝐊_𝐹𝐅^𝑇(𝑡),𝐊_𝐹=𝑘^𝐹_𝑚𝑛,𝐾()(𝑥)𝐊𝑚,𝑛=0,1,…,𝑁,𝑥,𝑡=𝐗_𝑡𝐗^𝑇(𝑡),𝐊_𝑡=𝑘^𝑡_𝑚𝑛,𝑚,𝑛=0,1,…,𝑁. (27)(28) From (11), (27), and (28) we can obtain 𝐗(𝑥)𝐊_𝑡𝐗^𝑇(𝑡)=𝐅(𝑥)𝐊_𝐹𝐅^𝑇(𝑡)⟹𝐗(𝑥)𝐊_𝑡𝐗^𝑇(𝑡)=𝐗(𝑥)𝐂^𝑇𝐊_𝐹𝐂𝐗^𝑇(𝑡). (29) Thus 𝐊_𝑡=𝐂^𝑇𝐊_𝐹𝐂⟹𝐊_𝐹=𝐂^𝑇−1𝐊_𝑡𝐂⁻¹. (30) By substituting the matrix forms (22) and (27) into the integral part 𝐼(𝑥) in (9), we can have the matrix relation as follows: [𝐼(𝑥)]=^𝑏_𝑎𝐅(𝑥)𝐊_𝐹𝐅^𝑇(𝑡)𝐗(𝑡)𝜷𝜇_𝑟,𝜏_𝑟𝐓^𝑇𝑘𝐂^𝑇(𝑥)𝐊𝐀𝑑𝑡=𝐅_𝐹𝐐𝐀 (31) so that 𝐐=^𝑏_𝑎𝐅^𝑇(𝑡)𝐗(𝑡)𝜷𝜇_𝑟,𝜏_𝑟𝐓^𝑇𝑘𝐂^𝑇𝑑𝑡. (32) From (5) and (32), we have 𝐐=^𝑏_𝑎𝐂𝐗^𝑇(𝑡)𝐗(𝑡)𝜷𝜇_𝑟,𝜏_𝑟𝐓^𝑇𝑘𝐂^𝑇𝜇𝑑𝑡=𝐂𝐇𝜷_𝑟,𝜏_𝑟𝐓^𝑇𝑘𝐂^𝑇, (33) where 𝐇=^𝑏_𝑎𝐗^𝑇(𝑡)𝐗(𝑡)𝑑𝑡,𝐇=ℎ_𝑖𝑗,ℎ_𝑖𝑗=𝑏^𝑖+𝑗+1−𝑎^𝑖+𝑗+1𝑖+𝑗+1𝑖,𝑗=1,2,…,𝑁. (34) If we substitute the matrix relation (5) into (31), we have the matrix form [𝐈(𝑥)]=𝐗(𝑥)𝐂^𝑇𝐊_𝐹𝐐𝐀. (35)

2.4. Matrix Relations for the Conditions

The corresponding matrix form for the conditions (2) can be shown, by means of (17), as ^𝑚−1_𝑘=0𝑎_𝑗𝑘𝐗(𝑎)+𝑏_𝑗𝑘𝐗(𝑏)𝐓^𝑇𝑘𝐂^𝑇𝐀=𝜆_𝑗,𝑗=1,2,…,𝑚. (36)

3. Method of Solution

We can construct the fundamental matrix equation corresponding for (1). For this aim, we substitute the matrix relations (23) and (35) into (9). So we obtain the matrix equation ^𝑚_𝑘=0𝑃_𝑘(𝑥)𝐗(𝑥)𝐂^𝑇𝐓^𝑇𝑘𝐀+^𝑠_𝑟=0𝑄_𝑟(𝑥)𝐗(𝑥)𝜷𝜇_𝑟,𝜏_𝑟𝐓^𝑇𝑟𝐂^𝑇𝐀=g(𝑥)+𝜆𝐗(𝑥)𝐂^𝑇𝐊_𝐹𝐐𝐀. (37) By using in (37) the collocation points 𝑥_𝑖 defined by 𝑥_𝑖=𝑎+𝑏−𝑎()𝑁−1𝑖−1,𝑖=1,2,…,𝑁, (38) the system of the matrix equations is obtained ^𝑚_𝑘=0𝑃_𝑘𝑥_𝑖𝐗𝑥_𝑖𝐂^𝑇𝐓^𝑇𝑘𝐀+^𝑠_𝑟=0𝑄_𝑟𝑥_𝑖𝐗𝑥_𝑖𝜷𝜇_𝑟,𝜏_𝑟𝐓^𝑇𝑟𝐂^𝑇𝐀=g𝑥_𝑖+𝜆𝐗𝑥_𝑖𝐂^𝑇𝐊_𝐹𝐐𝐀 (39) or shortly the fundamental matrix equation becomes ^𝑚_𝑘=0𝐏_𝑘𝐗𝐓^𝑇𝑘𝐂^𝑇+^𝑠_𝑟=0𝐐_𝑟𝐗𝜷𝜇_𝑟,𝜏_𝑟𝐓^𝑇𝑟𝐂^𝑇−𝜆𝐗𝐂^𝑇𝐊_𝐹𝐐𝐀=𝐆, (40) where 𝐏_𝑘=𝑝_𝑘𝑥₁0⋯00𝑝_𝑘𝑥₂⋯000⋱000⋯𝑝_𝑘𝑥_𝑁,𝐐_𝑟=𝑄_𝑟𝑥₁0⋯00𝑄_𝑟𝑥₂⋯000⋱000⋯𝑄_𝑟𝑥_𝑁,𝑋𝑥𝐗=¹𝑋𝑥₂⋮𝑋𝑥_𝑁=1𝑥₁⋯𝑥₁^𝑁−11𝑥₂⋯𝑥₂^𝑁−1⋮⋮⋱⋮1𝑥_𝑁⋯𝑥_𝑁^𝑁−1,g𝑥𝐆=¹g𝑥₂⋮g𝑥_𝑁. (41) Therefore, the fundamental matrix equation (40) corresponding for (1) can be written as 𝐖𝐀=𝐆or[𝐖;𝐆], (42) where 𝐖=^𝑚_𝑘=0𝐏_𝑘𝐗𝐓^𝑇𝑘𝐂^𝑇+^𝑠_𝑟=0𝐐_𝑟𝐗𝜷𝜇_𝑟,𝜏_𝑟𝐓^𝑇𝑟𝐂^𝑇−𝜆𝐗𝐂^𝑇𝐊_𝐹𝐐. (43) Equation (42) corresponds to a system of 𝑁 linear algebraic equations with unknown Fibonacci coefficients𝑎₁,𝑎₂,…,𝑎_𝑁. Further, we can express the matrix form (36) conditions 𝐔_𝑗𝐴=𝜆_𝑗or𝐔_𝑗;𝜆_𝑗,𝑗=1,2,…,𝑚, (44) where 𝐔_𝑗=^𝑚−1_𝑘=0𝑎_𝑗𝑘𝐗(𝑎)+𝑏_𝑗𝑘𝐗(𝑏)𝐓^𝑇𝑘𝐂^𝑇=𝑢_𝑗1𝑢_𝑗2𝑢_𝑗3…𝑢_𝑗𝑁. (45) To obtain the solution of (1) under the conditions (2), by replacing the row matrices (44) by the last 𝑚 rows of the matrices (42), we have the new augmented matrix 𝐖𝐀=𝐆. (46) If the last m rows of the (30) are replaced, the augmented matrix of the above system is obtained as 𝐆=𝑤𝐖;₁₁𝑤₁₂⋯𝑤_1𝑁;g𝑡₁𝑤₂₁𝑤₂₂⋯𝑤_2𝑁;g𝑡₂𝑤⋮⋮⋮⋮;⋮_{(𝑁−𝑚)1}𝑤_{(𝑁−𝑚)2}⋯𝑤_{(𝑁−𝑚)𝑁};g𝑡_𝑁−𝑚𝑢₁₁𝑢₁₂⋯𝑢_1𝑁;𝜆₁𝑢₂₁𝑢₂₂⋯𝑢_2𝑁;𝜆₂𝑢⋮⋮⋮⋮;⋮_𝑚1𝑢_𝑚2⋯𝑢_𝑚𝑁;𝜆_𝑚. (47) If rank 𝐖=rank[𝐖;𝐆]=𝑁, then we can write 𝐀=𝐖^−𝟏𝐆. (48) And so, the matrix 𝐀 (thereby the coefficients𝑎₁,𝑎₂,…,𝑎_𝑁) is uniquely determined.

4. Accuracy of Solution

We can check the accuracy of the method. The truncated Fibonacci series in (3) have to be approximately satisfying (1). For each 𝑥=𝑥_𝑖∈[𝑎,𝑏], 𝑖=1,2,3,…, 𝐸𝑥_𝑖=𝑃𝑥_𝑖−𝑄𝑥_𝑖−g𝑥_𝑖−𝜆𝐼𝑥_𝑖≅0 (49) or 𝐸𝑥_𝑖≤10^−𝑘𝑖𝑘_𝑖isanypositiveinteger. (50) If max(10^−𝑘𝑖)=10^−𝑘 (𝑘 is any positive integer) is prescribed, then the truncation limit 𝑁 is increased until the difference 𝐸(𝑥_𝑖) at each of the points 𝑥_𝑖 becomes smaller than the prescribed 10^−𝑘.

5. Numerical Examples

In this section, several examples are given to illustrate the applicability of the method and all of them are performed on the computer MATLAB. Also, the absolute errors in tables are the values of |𝑦(𝑥)−𝑦_𝑁(𝑥)| at selected points.

Example 1. Let us consider the linear Fredholm integro-differential-difference equation given by 𝑦^′′(𝑥)−𝑦^′′(𝑥−1)+2𝑥𝑦^′(𝑥−2)=4𝑥²−15𝑥+4+12¹₀𝑥𝑡𝑦(𝑡)𝑑𝑡,0≤𝑥,𝑡≤1, (51) with the boundary conditions 𝑦(0)=1,𝑦(1)=1 (52) and the approximate solution 𝑦(𝑥) by the truncated Fibonacci series 𝑦(𝑥)=³_𝑛=1𝑎_𝑛𝐹_𝑛(𝑥), (53) where 𝑁=3, 𝑃₂(𝑥)=1, 𝑄₁(𝑥)=2𝑥, 𝑄₂(𝑥)=1, 𝜇₁=1, 𝜏₁=−2, 𝜇₂=1, and 𝜏₂=−1, 𝜆=12,𝐾(𝑥,𝑡)=𝑥𝑡,g(𝑥)=4𝑥²−15𝑥+4. (54) From (38), the collocation points for 𝑁=3,are computed 𝑥₁=0,𝑥₂=12,𝑥₃=1 (55) and from (40), the fundamental matrix equation of the problem is 𝐏₂𝐗𝐓^𝑇2𝐂^𝑇+𝐐₀𝐗𝜷(1,−2)𝐓^𝑇0𝐂^𝑇+𝐐₁𝐗𝜷(1,−1)𝐓^𝑇2𝐂^𝑇−𝝀𝐗𝐂^𝑇𝐊_𝑓𝐐𝐀=𝐆, (56) where 𝐏₂=100010001,𝐐₁=000010002,𝐐₂=,𝐓100010001_𝑇=010002000,𝐂^𝑇=4−5101010001,𝐆=2,𝜷()=()=,11−71,−21−2401−4001,𝜷1,−11−1101−2000𝐊=000010000,𝐐=243121334433428.15 (57) The augmented matrix for this fundamental matrix equation is calculated as []=7𝐖;𝐆004;4−3−1−25;−2−6−2−9;−7. (58) From (37), the matrix forms for the boundary conditions are 𝐔₀;𝜆₀=,𝐔101;1₁;𝜆₁=012;1. (59) The new augmented matrix based on conditions can be written as 𝐖;𝐆=004;4101;1012;1. (60) Solving this system, the unknown Fibonacci coefficients are obtained as 𝐴=0−11^𝑇. (61) Hence, by substituting the Fibonacci coefficients matrix into (11), we have the approximate solution 𝑦(𝑥)=𝑥²−𝑥+1, which is the exact solution 𝑦(𝑥)=³_𝑛=1𝑎_𝑛𝐹_𝑛(𝑥)=𝑎₁𝐹₁(𝑥)+𝑎₂𝐹₂(𝑥)+𝑎₃𝐹₃(𝑥)=0.1+(−1)⋅𝑥+1⋅𝑥²+1=𝑥²−𝑥+1. (62)

Example 2 (see [26]). Consider the following linear Fredholm integro-differential-difference equation with variable coefficients 𝑦^′′′(𝑥)−𝑥𝑦^′(𝑥)+𝑦^′′()()(()(𝑥(2)𝑥−1−𝑥𝑦𝑥−1=−𝑥+1)(sin𝑥−1+cos))−cos+1+¹₋₁𝑦(𝑡−1)𝑑𝑡−1≤𝑥,𝑡≤1 (63) with the conditions 𝑦(0)=0, 𝑦^′(0)=1, and 𝑦^′′(0)=0 and the exact solution 𝑦(𝑥)=sin(𝑥).
So𝑃₁(𝑥)=−𝑥, 𝑃₃(𝑥)=1, 𝑄₁(𝑥)=2𝑥, 𝑄₂(𝑥)=1, 𝜇₁=1, 𝜏₁=−1, 𝜇₀=1, and 𝜏₀=−1,()g(𝑥)(()(𝑥(2)𝜆=1,𝐾𝑥,𝑡=𝑡−1,=−𝑥+1)(sin𝑥−1+cos))−cos+1. (64) The fundamental matrix equation of the problem from (40) becomes 𝐏₁𝐗𝐓^𝑇1𝐂^𝑇+𝐏₃𝐗𝐓^𝑇3𝐂^𝑇+𝐐₀𝐗𝜷(1,−1)𝐓^𝑇0𝐂^𝑇+𝐐₁𝐗𝜷(1,−1)𝐓^𝑇2𝐂^𝑇−𝝀𝐗𝐂^𝑇𝐊_𝑓𝐐𝐀=𝐆. (65)
For 𝑁=8,9 the solutions of the problem are found as, respectively, 𝑦(𝑥)=(0.003823057225)𝑥⁷+(0.01008064682)𝑥⁶−(0.02660425406)𝑥⁵+(0.05046207175)𝑥⁴+(0.1974487774)𝑥³−(2.925448512)𝑥²(),𝑦(𝑥)()𝑥+𝑥+5.629719781𝑒−17=−0.003384338243⁸−(0.001420180534)𝑥⁷+(0.06727516657)𝑥⁶−(0.1256152219)𝑥⁵−(0.2693617976)𝑥⁴+(2.000085149)𝑥³−(6.513886652𝑒−16)𝑥²+𝑥−(1.153157431𝑒−15). (66) We compare the solutions found by the present method for 𝑁=8, 𝑁=9 and the absolute errors in Figure 1 and Table 1. It is seen that when we increase integer 𝑁, the errors decrease.

Table 1 

Comparison of the absolute errors of Example 2.

Figure 1 

Numerical and exact solutions of Example 2 for 𝑁=8,9.

Example 3 (see [30]). Now consider the linear Fredholm integro-differential-difference equation given by 𝑦^′′′(𝑥)−(𝑥−1)𝑦^′′+(𝑥−1)𝑦^′(𝑥)−𝑦(𝑥)+𝑦^′(𝑥−1)=𝑒^𝑥−11+𝑥𝑒𝑥−𝑒1𝑥−2𝑒+¹₋₁𝑥𝑡−𝑥²𝑦(𝑡)𝑑𝑡−1≤𝑥,𝑡≤1 (67) with the initial conditions 𝑦(0)=0, 𝑦^′(0)=1, and 𝑦^′′(0)=0 and the exact solution 𝑦(𝑥)=𝑒^𝑥.
Here 𝑃₀(𝑥)𝑃=−1,₁(𝑥)𝑃=𝑥−1,₂(𝑥)=−(𝑥−1),𝑃₃(𝑥)𝑄=1,₁(𝑥)=1,𝜇₁=1,𝜏₁()=−1,𝜆=1,𝐾𝑥,𝑡=𝑥𝑡−𝑥²,g(𝑥)=𝑒^𝑥−11+𝑥𝑒𝑥−𝑒1𝑥−2𝑒. (68) From (40), the fundamental matrix equation of the problem becomes 𝐏₀𝐗𝐓^𝑇0𝐂^𝑇+𝐏₁𝐗𝐓^𝑇1𝐂^𝑇+𝐏₂𝐗𝐓^𝑇2𝐂^𝑇+𝐏₃𝐗𝐓^𝑇3𝐂^𝑇+𝐐₁𝐗𝜷(1,−1)𝐓^𝑇1𝐂^𝑇−𝝀𝐗𝐂^𝑇𝐊_𝑓𝐐𝐀=𝐆. (69) The solutions obtained for 𝑁=6,9 are compared with the exact solution is 𝑒^𝑥 which are given in Figure 2 and we compare the absolute errors found by present method for 𝑁=6,9 in Figure 3. Also, the numerical solution and absolute errors are compared for 𝑁=6,9 in Table 2.

Table 2 

Comparison of the absolute errors of Example 3.

Figure 2 

Numerical and exact solutions of Example 3 for 𝑁=6,9.

Figure 3 

Comparison of the absolute errors of Example 3 for 𝑁=6,9.

6. Conclusion

The Fibonacci collocation method is used to solve the linear Fredholm integro-differential-difference equations numerically. The obtained numerical results show that the accuracy improves when 𝑁 is increased. Tables and figures indicate that as 𝑁 increases, the errors decrease more rapidly. A considerable advantage of the method is that the Fibonacci polynomial coefficients of the solution are found very easily by using computer programs. This method can also be extended to the system of linear integro-differential-difference equation with variable coefficients, but some modifications are required.