New Variant of Arithmetic Mean Iterative Method for ...

2015 Third International Conference on Artificial Intelligence, Modelling and Simulation

A New Variant of Arithmetic Mean Iterative Method for Fourth Order Integro-differential Equations Solution E. Aruchunan Dept. of Mathematics and Statistics Curtin University Perth WA6845, Australia Email: [email protected]

Y. Wu Dept. of Mathematics and Statistics Curtin University Perth WA6845, Australia Email: [email protected]

B. Wiwatanapataphee Dept. of Mathematics and Statistics Curtin University Perth WA6845, Australia Email: [email protected]

P. Jitsangiam Dept. of Civil Engineering Curtin University Perth WA6845, Australia Email: [email protected]

where E invertible (N − 1) × (N − 1) matrix, f is a given vector, and ψ denotes the unknown vector to be determined. e we consider the Two-stage iterative method In this paper, (i.e. Arithmetic Mean (AM)) with some modification. The basic idea of the Two-stage iterative methods is to solve the outer iteration by using another iterative procedure called inner iteration. The concept of Two-stage iterative methods, which are also known as inner/outer iteration schemes have been applied successfully for solving various matrix problems. Based on the two-stage procedure, many iterative methods have been developed for solving linear systems such as Alternating Group Explicit (AGE) [3], Modified Alternating Group Explicit (MAGE) [4], Iterative Alternating Decomposition Explicit (IADE) [5], SOR [6], [7] and Weighted Mean (WM) [8], [9] methods. However, in this study, further discussions are focused on WM iterative methods. The family of WM iterative methods are broadly applied to a system of equations, and one such method is the Arithmetic Mean (AM) iterative method [8]. In a series of papers, the effectiveness of the AM method and its variants were studied and tested on linear and nonlinear systems that arise from various scientific problems [10], [11], [12], [13] and [17]. Nevertheless, all these methods are still based on a single optimal parameter which is used to accelerate the numerical computation and still requires a great deal of time to obtain the desired solutions. Therefore, in this study we introduce Modified Arithmetic Mean (MAM) to accelerate the numerical computation which is faster and more reliable than standard AM method and its variants. The effectiveness of the proposed method was also validated with some problems from fourth order Integro-differential equations. In Section 2, some theories

Abstract – In this paper, we introduced a new variant called Modified Arithmetic Mean iterative method (MAM) from standard Arithmetic Mean (AM)iterative method for the solution of fourth order Fredholm Integro-differential equations. The proposed method has been derived with some relevant theories and proof to validate the convergence. In addition, the formulation and implementation of the proposed method in solving some well-posed problems are also presented and investigated based on a number of iterations, CPU time and Root Square Mean Error (RSME). Numerical simulation and comparisons have been carried out and scrutinised with Gauss-Seidel and standard AM iterative methods to validate the effectiveness of the proposed method. Keywords – Arithmetic Mean; system of equations; Fredholm equations; Integro-differential equations.

I. I NTRODUCTION In real-life, many mathematical models of physical phenomena contain integro-differential equations (IDEs). These equations arise in biological models, nano-hydrodynamics, glass-forming processes and chemical kinetics; for more details see [1] and [2]. In genaral, to solve the IDEs, analytical methods are used to understand the mechanism and physical effects through the model’s problems. However, in solving complex problems physically and geometrically, analytical methods are impossible and time-consuming. Therefore, numerical methods are developed and implemented to enable us to obtain solutions of complex problems with the least mathematical operations. In this paper, we have introduced the Modified Arithmetic Mean (MAM) iterative method to review fast and efficient algorithms in solving dense matrix equation. Consider a system of algebraic equations Eψ = f e 978-1-4673-8675-3/15 $31.00 © 2015 IEEE DOI 10.1109/AIMS.2015.24

(1) 82

−U =  0 ζ2,1 0 0  0 0   .. .. . .  0 0  0 0 0 0

and formulations of the MAM method are given and in section 3, numerical approximation is derived. In section 4, some numerical experiments are carried out and the results are discussed briefly using percentage reduction analysis. In the last section, a conclusion is given. II. M ODIFIED A RITHMETIC M EAN I TERATIVE M ETHOD (MAM) A. Optimal Parameter Values Basically, accelerated parameter or optimal parameter value is used to accelerate the convergent rate of the methods[?]. Therefore, in this paper, we introduced a new optimal parameter value, θ2 which is used for backward iteration to accelerate the convergence. Therefore, there are two optimal parameter values used in MAM method which have improved the convergence rate compared to standard AM method. These optimal parameter values are determined by running the simulation for different values of θ1 and θ2 chosen based on the least number of iterations. In the case of θ1 = θ2 , our MAM method is equivalent to the standard AM method. In the MAM iterative method, each iteration consists of solving two independent systems such as ψ 1k+1 and ψ k+1 2 which can be developed by expressing E asefollows, e

τ1,N −3 τ2,N −3 τ3,N −3 .. .

τ1,N −2 τ2,N −2 τ3,N −2 .. .

0 0 0

··· ··· ···

0 0 0

ζN −2,N −2 0 0

ψ k+1 = TM AM ψ k2 + cM AM f e e

(2) TM AM = where,

 σ1,1  0   0   D =  ...   0   0 0

σ2,2 0 .. .

σ3,3 .. .

··· ··· ··· .. .

0 0 0

0 0 0

··· ··· ···

0

0 0

0 0 0 .. .

0 0 0 .. .

0 0 0 .. .

σx,x 0 0

0 σy,y 0

σz,z

ζ3,2 .. . τN −3,2 τN −2,2 τN −1,2

0 ··· 0 ··· 0 ··· .. . . . . 0 ··· 0 ··· 0 ···

0 0 0 .. .

0

0 0

ζN −2,N −3 χN −1,N −3

ζN −1,N −2

       χN −3,N −1   ζN −2,N −1  0

1 k+1 (ψ + ψ k+1 ) 2 2 e1 e

(4)

(5)

ψ k+1 = (D − θ1 L)−1 ((1 − θ1 )D − θ1 U ) 1 e k+1 ψ 2 = (D − θ2 U )−1 ((1 − θ2 )D − θ2 L) e and 1 cM AM = [θ1 (D − θ1 L)−1 + θ2 (D − θ2 U )−1 ] (6) 2 The general condition which guarantees the convergence of MAM iterative method for solving the linear system is described in the following theorems.

       0   0 

0 0 0 .. .





where σx,x = σN −3,N −3 σy,y = σN −2,N −2 σz,z = σN −1,N −1

0 0

τ1,N −1 τ2,N −1 τ3,N −1 .. .

 = ((1 − θ1 )D − θ1 U )ψ k1 + θ1 f  (D − θ1 L)ψ k+1 1     ek e k+1 (D − θ2 U )ψ 2 = ((1 − θ2 )D − θ2 L)ψ 2 + θ2 f  e e   1 k+1  k+1 k+1  ψ = (ψ 1 + ψ 2 ) 2 e e e (3) where (D − θ1 L) and (D − θ2 U ) are nonsingular matrices with acceleration parameters θ1 and θ2 . From (3), MAM can be expressed as follows

where,

and −L =  0  ζ2,1   χ3,1   ..  .  τN −3,1  τN −2,1 τN −1,1

··· ··· ··· .. .

Therefore, the general formulation for MAM method is defined as follows

whereas, E =D−L−U

χ1,3 0 0 .. .

Theorem 1 Let TM AM be (N − 1) × (N − 1) matrix and the successive approximation (4) for k = 0, 1, 2, · · · converge for each cM AM ⊆ R(N −1) . Each ψ (0) ∈ C N −1 if and only if the e matrix i.e TM AM is less than spectral radius of the iteration one, that is ρ(TM AM ) < 1. Proof. The standard proof is given in [16].

 0 0  0  ..  .  0  0 0

Theorem 2 A necessary condition for the MAM method to be convergent is that 0 < θ1 < 2 and 0 < θ2 < 2. The standard proof is given in [16].

and 83

Lemma 1 Since the eigenvalues, λ1 ,. . . ,λN of TM AM are the zero of the characteristic polynomial, thus determinants of the TM AM satisfy the following relations det(TM AM ) =

N Y

λj

subsequently for simplicity: ψ i ≡ ψ (xi ) ej ≡ e ψ ψ (tj ) fei ≡ fe(xi ) Ki,j ≡ K(xi , tj )

(7)

j=0

We apply the Central Difference and Composite Trapezoidal schemes to equation(1) and it reduces to

where multiple eigenvalues are repeated according to their algebraic multiplicity. Therefore, based on multiplication rules for determinants and since 12 (D − θ1 L), 12 (D − θ2 U ), 1 1 2 ((1 − θ1 )D − θ1 U ) and 2 ((1 − θ2 )D − θ2 L) are triangular matrices. Proof. For the convergence of MAM iterative method associated with the general conditions for matrix TM AM is described as follows

N X Ω1 + αi Ω2 + βi ψ i − Aj Ki,j ψ j e e j=0

(10)

for (i = 1, · · · , N − 1) where, ψ i+2 −4ψ i+1 +6i −4ψ i−1 +ψ i−2 Ω1 = e e e h4 e and ψ −2ψ +ψ i+1 i i−1 Ω2 = e , he2 e

det(TM AM ) =det[ 12 (D − θ1 L)−1 ((1 − θ1 )D − θ1 U ) + (D − θ2 L)−1 ((1 − θ2 )D − θ2 L)] =det[(D −θ1 L)−1 12 ((1−θ1 )D −θ1 U )+(D −θ2 L)−1 12 ((1− θ2 )D − θ2 L)] =det(D − θ1 L)−1 det 12 ((1 − θ1 )D − θ1 U ) + det(D − θ2 L)−1 det 21 ((1 − θ2 )D − θ2 L)] = 12 (1 − θ1 )N −1 + 12 (1 − θ2 )N −1 = 12 [(1 − θ1 )N −1 + (1 − θ2 )N −1 ]

meanwhile, the quadrature weights, Aj satisfies the following relations  Aj =

1 2 h,

h,

j = 0, N . otherwise

(11)

This now implies 1 1 (1 − θ1 ) + (1 − θ2 ) | . (8) 2 2 The standard proof is given in [17]. From the Theorem 1, concluded that the necessity of 0 < θ1 < 2 and 0 < θ2 < 2 for convergence. ρ(TM AM ) ≥|

which is can be expressed in the matrix form as follows Eϕ = f e

III. N UMERICAL A PPROXIMATION OF I NTEGRO -D IFFERENTIAL E QUATION

(12)

E=

 · · · τ1,N −3 τ1,N −2 τ1,N −1 · · · τ2,N −3 τ2,N −2 τ2,N −1    · · · τ3,N −3 τ3,N −2 τ3,N −1     · · · τ4,N −3 τ4,N −2 τ4,N −1    .. .. .. ..     . . . . Z b 4 2  d ψ d ψ τN −4,1 τN −4,2 τN −4,3 · · · ζN −4,N −3 χN −4,N −2 τN −4,N −1   K(x, t)ψ (t)dt = f (x), x ∈ (a, b) τN −3,1 τN −3,2 τN −3,3 · · · σN −3,N −3 ζN −3,N −2 χN −3,N −1  e (x)+α e2 (x)+βψ −  dx4 dx   e a e τ τ τ · · · ζ σ ζ N −2,1 N −2,2 N −2,3 N −2,N −3 N −2,N −2 N −2,N −1 (9) τN −1,1 τN −1,2 τN −1,3 · · · χN −1,N −3 ζN −1,N −2 σN −1,N −1 with boundary conditions, σ1,1 ζ2,1 χ3,1 τ4,1 .. .

ζ1,2 σ2,2 ζ3,2 χ4,2 .. .

χ1,3 ζ2,3 σ3,3 ζ4,3 .. .

In this paper, we test the developed MAM iterative method through the fourth order Fredholm Integro-differential equations (FIDE) as follows



ψ (a) = a0 , ψ (b) = b0 , ψ 00 (a) = a2 and ψ 00 (b) = b2 . e e e e K(x, t), f (x), α, β, a0 , a2 , b0 and b2 are known functions whereas ψ (x) is the unknown function to be determined. Let thee interval (a, b) be divided uniformly into (N − 1) subintervals and the discrete set of points of tj , (j = 0, 1, · · · , N − 1, N ) , h is the constant step size defined as h = b−1 N . The following notation will be used

whereas, E is order of (N − 1) × (N − 1), with σ1,1 = 1 + σ1,1 and σN −1,N −1 = 1 + σN −1,N −1 σi,i = 6 − 2h2 αi + h2 βi − h4 Ai Ki,i ζi,j = −4 + h2 αi − h4 Aj Ki,j χi,j = 1 − h4 Aj Ki,j τi,j = −h4 Aj Ki,j

84

Problems − A[18] FIDE − A =:  4 R1 d ψ    dxe4 (x) − ψ (x) − x(1 + ex ) − 3ex + 0 ψ (t)dt = 0, e e x ∈ (1, 0)   ψ (0) = 1, ψ (1) = 1 + e, ψ 00 (0) = 2, ψ 00 (1) = 3e e e e e (13)

f=   4 h f1 + h2 a2 + (6 − h2 α1 − τ1,0 )y0 − (τ1,N )yN 4 h f2 + (−1 − (τ2,0 )y0 − (τ2,N )yN     h4 f3 − (τ3,0 )y0 − (τ3,N )yN   4   h f − (τ )y − (τ )y 2 2,0 0 2,N N     . ..     4   h fN −4 − (τN −4,0 )y0 − (τN −4,N )yN   4   h fN −3 − (τN −3,0 )y0 − (τN −3,N )yN   4 h fN −2 − (τN −2,0 )y0 − (−1 − τN −2,N )yN 4 2 h fN −1 + h b2 − (τN −1,0 )y0 + ∆

and the exact solution is ψ (x) = 1 + xex . e Problems − B[19] FIDE − B =:  4 R1 d2 ψ d ψ    dxe4 − 2 dxe2 − 3x − 0 xψ (t)dt = 0, e x ∈ (1, 0)   ψ (0) = 0, ψ (0) = 1, ψ 00 (0) = 0, ψ 00 (1) = e e e e 3 and the exact solution is ψ (x) = 87 x + 61 47 47 x. e

where, ∆ = (6 − h2 αN −1 − τN −1,N )yN and 



ϕ(x1 )  ϕ(x2 )     ϕ(x3 )     ϕ(x4 )      .. ϕ=  .   e   ϕ(xN −4 ) ϕ(xN −3 )   ϕ(xN −2 ) ϕ(xN −1 )

(14) 87 47

Considering problems FIDE-A and FIDE-B, the best optimal parameter values are used for the fast convergence recorded in Tables 1(a) and 2(a) to obtain the minimum number of iterations, which are illustrated as in the Figures 1 and 3. Figures show that the proposed method required the least number of iterations compared to the other two methods. Meanwhile, Table 1(b) and 2(b) show the CPU time required for each iterative method to yield the solutions in seconds. From Figures 2 and 4, the MAM iterative method converged faster than the GS and AM iterative methods. Table 1(c) and 2(c) display the Root Mean Square Error of each method associated with each mesh size and the improvement of the results. The results show that the accuracy of the proposed method is comparable with other two methods.

Obviously, E is a nonsymmetric dense matrix with the dimension of order with (N − 1) by (N − 1). Now then, the system (11) is solved by GS, AM and MAM iterative methods. IV. N UMERICAL E XPERIMENTS Effectiveness of the MAM to the standard AM and GS is investigated numerically through two problems of fourth order FIDE-A and FIDE-B. All numerical experiments are performed on a personal computer with Intel(R) Core(TM) i5-4460 CPU @ 3.20GHz and 8.0GB RAM. We developed mathematical coding based on C compiler and performed numerical experiments to investigate the effectiveness of the MAM method using two optimal parameters values θ1 and θ2 . The new proposed optimal parameter value,θ2 is determined based on the best θ1 obtained from the standard AM iterative method. We chose three criteria, including a number of iterations, CPU time and Root Mean Square Error (RMSE), to compare the effectiveness of three methods associated with several mesh sizes. The numerical results of the tested methods, GS, AM and MAM are tabulated in Table 1 and 2. Throughout the simulations, the convergence criteria were reached when the tolerance error less than,  = 1010 .

TABLE I C OMPARISON OF A NUMBER OF ITERATIONS , CPU TIME AND ROOT MEAN SQUARE ERROR FOR THE ITERATIVE METHODS AT OPTIMAL PARAMETER VALUES OF θ1 = 1.8 AND θ2 = 1.9

Methods GS AM MAM

24 33008 8775 6830

Methods GS AM MAM

24 1.34 0.53 0.35

Methods GS AM MAM

85

24 6.689E-4 3.151E-4 3.151E-4

(a)Number of iterations Mesh Sizes, N 48 72 96 434104 1931044 5511073 118961 537239 1551700 82333 370969 1068565 (b)CPU time (seconds) Mesh Sizes, N 48 72 96 22.08 179.02 535.77 8.36 62.93 285.63 5.71 43.01 190.31 (c)Root Mean Square Error Mesh Sizes, N 48 72 96 4.419E-5 2.442E-5 1.531E-5 3.299E-5 3.335E-5 3.350E-5 3.299E-5 3.335E-5 3.350E-5

120 12337990 3511122 2411659 120 1469.56 940.52 614.49 120 1.062E-5 3.373E-5 3.373E-5

TABLE II C OMPARISON OF A NUMBER OF ITERATIONS , CPU TIME AND ROOT MEAN SQUARE ERROR FOR THE ITERATIVE METHODS AT OPTIMAL PARAMETER VALUES OF θ1 = 1.8 AND θ2 = 1.9

Methods GS AM MAM

24 29387 7845 5404

Methods GS AM MAM

24 1.35 0.50 0.36

Methods GS AM MAM

24 4.890E-5 4.881E-5 4.881E-5

(a)Number of iterations Mesh Sizes, N 48 72 96 380320 1670949 4716312 105004 469568 1344611 72743 324215 925126 (b)CPU time (seconds) Mesh Sizes, N 48 72 96 24.37 169.88 644.08 7.69 54.97 249.00 5.48 38.52 177.22 (c)Root Mean Square Error Mesh Sizes, N 48 72 96 1.361E-5 1.677E-5 4.872E-5 1.255E-5 7.440E-6 1.272E-6 1.255E-5 7.440E-6 1.272E-6

120 10449275 3018433 2069921 120 2223.63 746.44 534.66

Figure 3. Number of iterations versus mesh size of the GS, AM and MAM methods of FIDE-B

120 1.186E-6 2.972E-6 2.972E-6

Figure 4. CPU time versus mesh size of the GS, AM and MAM of FIDE-B

Figure 1. Number of iterations versus mesh size of the GS, AM and MAM methods of FIDE-A

Tables 3 and 4 show the percentage reduction of the MAM in terms of the number of iterations and CPU time from problem A and B respectively. It clearly shows that the number of iterations of the proposed MAM method has decreased by approximately 80% and 30% compared to GS and AM methods respectively. The CPU time of the MAM method has also been reduced by approximately 70% and 30% compared to GS and AM methods respectively. TABLE III P ERCENTAGE REDUCTION IN NUMBER OF ITERATIONS AND CPU TIME FOR MAM METHOD COMPARED TO THE GS AND AM METHODS FOR PROBLEM A

Methods GS AM

Figure 2. CPU time versus mesh size of the GS, AM and MAM of FIDE-A

24 79.31 22.16

Methods GS AM

86

24 73.88 33.96

(a)Number of iterations,% Mesh Sizes, N 48 72 96 81.03 80.78 80.61 30.78 30.94 31.14 (b)CPU time (seconds),% Mesh Sizes, N 48 72 96 74.13 75.97 68.05 31.69 31.65 33.37

120 80.45 31.31 120 68.95 34.66

TABLE IV P ERCENTAGE REDUCTION IN NUMBER OF ITERATIONS AND CPU TIME FOR MAM METHOD COMPARED TO THE GS AND AM METHODS FOR PROBLEM B

Methods GS AM

24 81.61 31.11

Methods GS AM

24 73.33 28.01

(a)Number of iterations,% Mesh Sizes, N 48 72 96 80.87 80.59 80.38 30.72 30.95 31.19 (b)CPU time (seconds),% Mesh Sizes, N 48 72 96 77.51 77.32 72.49 28.73 29.92 28.82

[8] V. Ruggiero, and E. Galligani,” An iterative method for large sparse systems on a vector computer,” Computers and Mathematics with Applications, vol. 20, pp. 25-28, 1990. [9] J. Sulaiman, M. Othman, N. Yaacob, and M. K. Hasan, ”HalfSweep Geometric Mean (HSGM) method using fourth-order finite difference scheme for two-point boundary problems,” Proceedings of the First International Conference on Mathematics and Statistics, June 19-21, 2006, Bandung, Indonesia, pp. 25-33, 2006. [10] M. Benzi, and T. Dayar, ”The Arithmetic Mean method for finding the stationary vector of Markov chains,” International Journal of Parallel, Emergent and Distributed Systems, 6(1): pp. 25-37, 1995. [11] I. Galligani, and V. Ruggiero,”The Arithmetic Mean method for solving essentially positive systems on a vector computer,” International Journal of Computer Mathematics, vil. 32, pp. 113121, 1990. [12] I. Galligani, and V. Ruggiero, ”The two-stage Arithmetic Mean method. Applied Mathematics and Computation, vol. 85, pp. 245264. 1997. [13] M. S. Muthuvalu and J. Sulaiman, ”Half-Sweep Arithmetic Mean method with composite trapezoidal scheme for solving linear Fredholm integral equations,” Appl. Math. Comput., vol. 217, pp. 5442-5448, 2011. [14] M. S. Muthuvalu, E. Aruchuan and J. Sulaiman, ”Solving first kind linear Fredholm integral equations with semi-smooth kernel using 2-point half-sweep block arithmetic mean method,” AIP Conference Proceedings, vol. 1557, pp. 350-354, 2013. [15] A. Hadjidimos, ”Successive overrelaxation (SOR) and related methods,” Journal of Computational and Applied Mathematics, vol. 123, pp. 177-199, 2000. [16] R. Kress, Numerical Analysis, New York: Springer, 1998. [17] M. S. Muthuvalu and J. Sulaiman, ”The Quarter-Sweep Geometric Mean Method for Solving Second Kind Linear Fredholm Integral Equations,” Bull. Malays. Math. Sci. Soc. , vol 36, pp. 10091026, 2013. [18] N. H. Sweilam, ”Fourth order integro-differential equations using variational iteration method,”. Computers and Mathematics with Applications,vol. 54 pp. 1086-1091, 2007. [19] J. Zhao and R. M. Corless, ”Compact finite difference method for integro-differential equations,” Applied Mathematics and Computation. vol. 177, pp. 271-288,2006.

120 80.19 31.42 120 75.96 28.37

V. C ONCLUSION In this paper, we introduced a new modified AM iterative method to solve dense linear systems arising from finite difference and composite trapezoidal approximation of the fourth order linear FIDE. Throughout the formulation and implementation, it is clearly shown that the MAM iterative method has reduced the number of iterations and CPU time significantly compared to the GS and the standard AM iterative method. The accuracy of the proposed MAM iterative methods is also in agreement. ACKNOWLEDGMENT The author would like to thank International Postgraduate Research Scholarship (IPRS) for the financial support and Dr Jumat Sulaiman and Dr Roger Collinson for their constructive comments to improve the quality of the paper. R EFERENCES [1] S. Yalcinbas and M. Sezer, ”The approximate solution of highorder linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials,”. Applied Mathematics and Computation, vol. 112, pp 291-308. 2000. [2] E. Aruchunan, M. S.Muthuvalu and J. Sulaiman, ”QuarterSweep Iteration Concept on Conjugate Gradient Normal Residual Method via Second Order Quadrature-Finite Difference Schemes for Solving Fredholm Integro-Differential Equation,” Sains Malaysiana, vol. 44, pp. 139-146, 2015. [3] D. J.Evans, ”The Alternating Group Explicit (AGE) matrix iterative method,” Applied Mathematical Modelling, vol. 11, pp 256-263, 1987. [4] D. J.Evans and W. S.Yousif, ”The Modified Alternating Group Explicit (M.A.G.E.) method,” Applied Mathematical Modelling, vol. 12, pp. 262-267, 1988. [5] M. S.Sahimi, A. Ahmad and A. A.Bakar, ”The Iterative Alternating Decomposition Explicit (IADE) method to solve the heat conduction equation,” International Journal of Computer Mathematics, vol. 47, pp. 219-229, 1993. [6] F. Cai, J. Xiao and Z. H.Xiang, ”Block SOR two-stage iterative methods for solution of symmetric positive definite linear systems,” Proceedings of the 3rd International Conference on Advanced Computer Theory and Engineering, August 20-22, 2010, Chengdu, China. 378-382.2010 [7] M. S. Muthuvalu, E. Aruchunan, J. Sulaiman, S. A. A.Karim and M. M. Rashidi, ”The 2-Point Explicit Group Successive OverRelaxation Method for Solving Fredholm Integral Equations of the Second Kind,” Applied Numerical Mathematics and Scientific Computation, pp. 67-70, 2013.

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