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In this paper, a modified algorithm is proposed for solving linear integro-differential equations of the second kind. The main idea is based on applying Romberg extrapolation algorithm (REA), on Trapezoidal rule. In accordance with the computational perspective, the comparison has shown that Adomian decomposition approach is more effective to be utilized. The numerical results show that the modified algorithm has been successfully applied to the linear integro-differential equations and the comparisons with some existing methods appeared in the literature reveal that the modified algorithm is more accurate and convenient.

Mathematical modelling of real-life problems usually results in functional equations, such as differential, integral, and integro-differential equations. Many mathematical formulations of physical phenomena reduced to integro-differen- tial equations, like fluid dynamics, biological models and chemical kinetics [

The numerical solution of integro-differential equation is a part of numerical analysis, which has been changed by the ongoing revolution in numerical methods. With the development of technology, useful methods are evolving for full utilization of the inherent powers of high speed and large memory computing machine. Many significant methods were discovered to approximate the solution of linear integro-differential equations, such as the Improved Bessel collocation method [

Without loss of generality, the linear Fredholmintegro differential equation of the second kind was considered.

u ′ ( x ) = f ( x ) + ∫ a b k ( x , t ) u ( t ) d t , u ( a ) = α , (1)

where, α is a real constant.

Some authors studied the numerical solution of the nonlinear integro-differ- ential equations by the Adomian decomposition method and compared it with the variational iteration method [

Saadati et al. [

In [

M. T. Rashed [

Jaradat et al. [

Mostafa Nadir and Azedine Rahmoune [

The paper is organized as follows: In Section 2, the proposed technique for solving the linear integro-differential equations is introduced. Some numerical experiments are presented in Section 3. The paper is concluded in Section 4.

In this section, the chosen algorithm is introduced for the solution of Fredholm integro-differential equation of the second kind. First, the Trapezoidal rule is applied to approximate the integral and the finite difference to approximate the derivative in (1), and then Romberg extrapolation is applied to increase the accuracy of the solution.

The numerical setting and the approximation of the integral for the Volterra Equation (1) will result in the coefficient matrix of the linear system of equations being a lower triangular one, which is exactly due to the variable upper limit x of the integration in (1), because in this equation the kernel k ( x , t ) ≡ 0 for t > x as the integrand can be considered identically zero above it is upper limit of integration x. So, for the discrete case k ( x i , t j ) = k i j = 0 for j > i is used. Then the system of linear equations with such a natural triangular coefficient matrix can be solved easily.

To use Trapezoidal rule the interval of integration ( a , x ) has been partitioned into n equally spaced subintervals of width

h = x n − a n , n ≥ 1 , (2)

where x n is the end point chosen for x; set t_{0} = a should be set and

t j = a + j h = t 0 + j h , j = 1 , 2 , 3 , ⋯ , n . (3)

Then the approximation of the integral in the integro-differential Equation (1) is given by,

∫ a x k ( x , t ) u ( t ) d t ≅ h [ 1 2 k ( x , t 0 ) u ( t 0 ) + k ( x , t 1 ) u ( t 1 ) + ⋯ + k ( x , t n − 1 ) u ( t n − 1 ) + 1 2 k ( x , t n ) u ( t n ) ] , (4)

where,

h = t j − a j = x − a n , t j ≤ x , j ≥ 1 , x = x n = t n . (5)

Replace n by 2 n = m , then (4) becomes,

∫ a x k ( x , t ) u ( t ) d t ≅ h [ 1 2 k ( x , t 0 ) u ( t 0 ) + k ( x , t 1 ) u ( t 1 ) + ⋯ + k ( x , t m − 1 ) u ( t m − 1 ) + 1 2 k ( x , t m ) u t m ] , (6)

which is called R(n, 0). Then Romberge algorithm (REA) R_{k}_{,j} can be applied,

where,

R k , j = R k , j − 1 + R k , j − 1 − R k − 1 , j − 1 4 j − 1 , i , j ≥ 1 (7)

Now, substitute Equation (6) in (1), thus obtain,

u ′ ( x ) = f ( x ) + h [ 1 2 k ( x , t 0 ) u ( t 0 ) + k ( x , t 1 ) u ( t 1 ) + ⋯ + k ( x , t m − 1 ) u ( t m − 1 ) + 1 2 k ( x , t m ) u ( t m ) ] , (8)

If n values of u ′ i = u ′ ( x i ) = u ′ ( t i ) are considered and,

k ( x , t j ) u ( t j ) = k ( x i , t j ) u ( t j ) , i = 1 , 2 , 3 , ⋯ , n ,

then Equation (8) becomes,

u ′ ( x i ) = f ( x i ) + h [ k ( x i , t 0 ) u ( t 0 ) 2 + k ( x i , t m ) u ( t m ) 2 + k ( x i , t 1 ) u ( t 1 ) + ⋯ + k ( x i , t m − 1 ) u ( t m − 1 ) ] (9)

For simplicity Equation (9) becomes,

u ′ i = f i + h [ k i 0 u 0 2 + k i , m u m 2 + k i 1 u 1 + ⋯ + k i , m − 1 u m − 1 ] . (10)

The finite difference formula is applied,

u ′ ( x ) ≅ 3 u ( x ) − 4 u ( x − h ) + u ( x − 2 h ) 2 h

to approximate u ′ i ( x ) in Equation (10) to get the following equation

3 u m − 4 u m − 1 + u m − 2 2 h = f i + h [ k i 0 u 0 2 + k i , m u m 2 + k i 1 u 1 + ⋯ + k i , m − 1 u m − 1 ] . (11)

In matrix notation, Equation (11) transform into the following system of linear equations:

K U = F ,

where,

K = [ ( − ) h 2 k 11 ( 1 − 2 h 2 k 12 ) ( − 2 h 2 k 13 ) ⋯ − h 2 k 1 m − ( 2 h 2 k 21 + 1 ) − 2 h 2 k 22 ( 1 − 2 h 2 k 23 ) ( 1 − 2 h 2 k 23 ) ( 1 − 2 h 2 k 23 ) ⋯ − h 2 k 12 m − 2 h 2 k 31 − ( 2 h 2 k 32 + 1 ) − h 2 k 33 ⋮ − 2 h 2 k m − 1 , 1 − 2 h 2 k m − 1 , 2 ⋯ − 2 h 2 k m − 1 , m − 1 ( 1 − 2 h 2 k m − 1 , m ) − 2 h 2 k m , 1 − 2 h 2 k m , 2 ⋯ − ( 2 h 2 k m , m − 1 + 4 ) − ( 2 h 2 k m , m − 3 ) ] (13)

U = ( u 1 , u 2 , ⋯ , u m − 1 , u m ) t (14)

And,

F = [ 2 h f 1 + ( h 2 k 10 + 1 ) u 0 , 2 h f 2 + ( h 2 k 20 ) u 0 , ⋯ , 2 h f m − 1 + h 2 k m − 1 , 0 ( u 0 ) , 2 h f m + h 2 k m − 1 , 0 ( u 0 ) ] t . (15)

This system can be solved for the unknowns u_{i}’s, i = 1 , 2 , ⋯ , m easily.

In this paper, illustrative examples of integro-differential equation are given to demonstrate the accuracy and efficiency of the proposed technique and compare it with some other existing methods.

Example 3.1 [

u ′ ( x ) = 1 − 1 3 x + ∫ 0 1 x t u ( t ) , u ( 0 ) = 0 , (16)

In Equation (16),

f ( x ) = 1 − 1 3 x , k ( x , t ) = x t .

The exact solution of Equation (3.1) is:

u ( x ) = x . (17)

The modified algorithm of Romberg extrapolation is applied for solving this example. Following equation is used:

h = h 5 = 1 16 , then k = 5 , with x 0 = 0 , and x 16 = 1 , with mesh points,

x i = i h 5 , i = 1 , 2 , ⋯ , 16. (18)

Apply Equation (10) and Equation (11) to compute the approximate solution,

u i , i = 1 , 2 , ⋯ , 16.

_{5,2}. It is shown that the proposed algorithm is accurate and efficient. Based on the recursive relations of Romberg extrapolation the accuracy increased with less computation time.

In [

However, the technique has the advantage, on the time R_{k}_{,1} is computed, the accuracy will increase with less computation as the recursive relation of Romberg increases.

Example 3.2 [

x_{i} | u_{i} using Romberg R_{5,2} | Exact value of u(x_{i}) | Absolute error |
---|---|---|---|

0.0625 | 0.0625015 | 0.0625 | |

0.125 | 0.125 | 0.125 | 0 |

0.1875 | 0.187513 | 0.1875 | |

0.25 | 0.250005 | 0.25 | |

0.3125 | 0.312536 | 0.3125 | |

0.375 | 0.375662 | 0.375 | |

0.4375 | 0.437571 | 0.4375 | |

0.5 | 0.500022 | 0.5 | |

0.5625 | 0.562618 | 0.5625 | |

0.625 | 0.625684 | 0.625 | |

0.6875 | 0.687676 | 0.6875 | |

0.75 | 0.750084 | 0.75 | |

0.8125 | 0.812746 | 0.8125 | |

0.875 | 0.875717 | 0.875 | |

0.9375 | 0.937827 | 0.9375 | |

1 | 1.00008 | 1 |

The method | Norm of the absolute errors |
---|---|

REA | |

CAS Wavelet | |

DTM | |

ESA |

u ′ ( x ) = 1 + sin x + ∫ 0 x u ( t ) , u ( 0 ) = − 1 , (19)

In Equation (19), f ( x ) = 1 + sin x , k ( x , t ) = 1. The exact solution of Equation (3.3) is

u ( x ) = 1 4 e x − 3 4 e − x − 1 2 cos x . (20)

The modified algorithm of Romberg extrapolation is applied for solving this example.

The study use h 5 = 1 16 , then, k = 5 , with x 0 = 0 , and x 16 = 1 , with mesh

points,

x i = i h 5 , i = 1 , 2 , ⋯ , 16.

Apply Equation (11) to compute the approximate solution u i , i = 1 , 2 , ⋯ , 16.

_{5,2}. It is shown that the proposed algorithm is accurate and efficient. Based on the recursive relations of Romberg extrapolation the accuracy increased with less computation

time.

In [

A modified technique by using Romberg extrapolation on the Trapezoidal rule was introduced to find an approximate solution of the linear integro-differential equation. Some numerical examples appearing in the literature are presented for introducing the main idea behind the approach and for comparisons purposes. The numerical results show that the technique has been successfully applied to the linear integro-differential equations with first derivative. Comparisons with the methods provided in [

x_{i} | u_{i} using Romberg R_{5,2} | Exact value of u(x_{i}) | Absolute error |
---|---|---|---|

0.0625 | −0.939313 | −0.93746 | |

0.125 | −0.877167 | −0.874684 | |

0.1875 | −0.816836 | −0.811451 | |

0.25 | −0.752159 | −0.74755 | |

0.3125 | −0.691295 | −0.682786 | |

0.375 | −0.623507 | −0.616973 | |

0.4375 | −0.561194 | −0.549936 | |

0.5 | −0.489657 | −0.481509 | |

0.5625 | −0.425194 | −0.411536 | |

0.625 | −0.349485 | −0.339866 | |

0.6875 | −0.282077 | −0.266357 | |

0.75 | −0.201611 | −0.190869 | |

0.8125 | −0.130719 | −0.11327 | |

0.875 | −0.0451471 | −0.0334261 | |

0.9375 | −0.0299532 | −0.0487906 | |

1 | −0.121113 | −0.13351 |

The method | Norm of the absolute errors |
---|---|

REA | |

CAS Wavelet | |

DTM | |

ESA |

convenient than the other methods. When the technique is compared with the VIM, the VIM normally gives a better accuracy than the method selected. However, the technique has one advantage: the accuracy can be increased with less computation as the recursive relations of Romberg increases.

The authors are very thankful to all the associated personnel in any reference that contributed in/for the purpose of this research. Further, this research holds no conflict of interest and is not funded through any source.

Al-Towaiq, M. and Kasasbeh, A. (2017) Modified Algorithm for Solving Linear Integro-Differential Equations of the Second Kind. American Journal of Computational Mathematics, 7, 157-165. https://doi.org/10.4236/ajcm.2017.72014