Microsoft Word - 178-186   178 Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1918 Vol. 32 (1) 2019 A parallel Numerical Algorithm for Solving Some Fractional Integral Equations Khalid Mindeel Mohammed Al-Abrahemee Khalid.mohammed@qu.edu.iq Department of Mathematics, College of Education,University of AL-Qadisiyhah, Qadisiyhah,Iraq. Article history: Received 12 August 2018, Accepted 26 September 2018, Publish January 2019 Abstract In this study, He's parallel numerical algorithm by neural network is applied to type of integration of fractional equations is Abel’s integral equations of the 1st and 2nd kinds. Using a Levenberge – Marquaradt training algorithm as a tool to train the network. To show the efficiency of the method, some type of Abel’s integral equations is solved as numerical examples. Numerical results show that the new method is very efficient problems with high accuracy. Keywords: Neural network, Abel’s integral equations, Levenberg – Marquardt training algorithm. 1. Introduction Abel’s equations are interconnected to a wide-ranging of physecal problems, such as tempera- ture transfer [1], nonlinear diffusion [2], the propagation of nonlinear waves [3], and were used in the theory of nautron transportation and traffic theory. There is several methods counting numerical analysis thus far away to reviewing this type of integration as well as their variations with several uses [4–5]. In [6] he also talked about solutions to this type of integration under assured conditions by some special ways. Sumner calculated this type from the point of observation of the convolutional convert studied a numerical solution for this integration using orthogonal polynomials. Hilbert worked on the problem of nonlinear type of energy, explained in locked shaped by expressive a sectional holomorphic function by earnings of an integral with energy nucleus, and converted the problem to one of resolving a generalized reviewing this type of integration. In [7] used the reverse of several of fractional order. In [8] resolved this integration numerically It is based on the process of approximation of partial integrations and derivatives. In [9] the researcher used Chebyshev polynomials for solve this integration. In [10] for easy calculations the researcher used some conversions such homotopy pertuarbation and Laplace transforms algorithm. 2. Abel’s Integral Equations This type of integration studied by Nieal Henreik Abel and Liouveille for its real importance in modeling many phenomena in mechanical, electronic, engineering, chemical and basic sciences [11- 13]. Where it emerged for this type of integration two types first and second as follows respectably   179 Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1918 Vol. 32 (1) 2019 𝑓 𝑥 𝜆 𝑦 𝑡 𝑥 𝑡 𝑑𝑡 1 And 𝑦 𝑥 𝑓 𝑥 𝜆 𝑑𝑡 2 Where 𝜆, 𝛼 𝑎𝑛𝑑 𝑓 𝑥 and the function are given where 0 𝛼 1, 𝑓 𝑐 ∈ 𝑐 0,1 , 0 𝑥, 𝑡 𝑇 and T is constant. We will solve this type of integration of fractional equations by means of neural networks using three layers: the input and output layers and the hidden layer contain 7 hidden units and one linear output unit, the sigmoid activation function of each unit in hidden layer is tanseg. function by a type of training, the Levenberge – Marqaurdt training algoreithm is used to train the network. 3. Description of the Method In this part we will describe how our method can be used to discovery the approximate result of Equations (1) and (2) let 𝛾 𝜒 denotes the result to be calculated, 𝛾 𝜒 , 𝜌 ) refers to the analytical solution. In the proposed approach, the FFNN experimental solution is used and the factors 𝜌 correspond to the weights and bias of the neural architecture. We choose a model for a pilot function yt (x) to meet BC requirements. This is attained by lettering it as two sets: 𝛾 𝜒 , 𝜌 𝜙 𝜒 𝜉 𝜒, Ν 𝜒, 𝜌 (3) where Ν 𝜒, 𝜌 is a singleoutput FFNN with 𝜌 factors and one input element breast-feeding on the 𝜒 input vector. The part 𝜙 𝜒 does not contain parameters that can be adjusted and meet the boundary conditions. The second term 𝜉 𝜒, Ν 𝜒, 𝜌 is created so that it does not contribute to BC, because 𝛾 𝜒 satisfies them. This part can be molded using FFNN whose weights and biases must be adjusted to address the minimization problem. 4. Illustration of the Method To explain the method, Let the equations ((1) and (2)), where x ∈ [0, 1] and the BC: y(0) = Α and y(1) = 𝛽, the approximation of solution can be written as: 𝛾 𝜒 , 𝜌 Α 𝛽 Α 𝜒 𝜒 𝜒 1 Ν 𝜒, 𝜌 (4) The error calculated to be minimized is given by 𝐸 𝑝 ∑ 𝑓 𝛾 , 𝛾 , 𝜒 𝑑𝑡 (5)   180 Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1918 Vol. 32 (1) 2019 where the xi’s are points in [a, b]. 5. Numerical Examples In this section we will apply the above algorithm to three different examples of the first kind and second kind. For each test problem the analytic solution 𝛾 𝜒 was identified in improvement. So, we have tried the accuracy of the solutions obtained through the equation; ∆𝑦 𝑥 |𝛾 𝜒 𝛾 𝜒 |" (6) Example 1 In the first example we will demonstrate the work of the proposed algorithm on the first type of fractional equations for that integration [14] ∪ 𝑑𝑡 , 0 1, where the exact solution ∪ 𝑡 Table 1. Analytic and Neural solution of example 1 The error E(x)  | yt(x) ya(x) | w h e r e yt(x) computed by the following training algorithm ( Trainlm) Out of suggested FFNN yt(x) for LM training algorithm Analytic solution input Trainlmya(x) x 0.000000000000000.000000000000000.00000000000000 0.0 0.000000022894650.1111111340057650.11111111111111 0.1 0.000763781650320.222986003872540.22222222222222 0.2 0.000000000654060.333333333987390.33333333333333 0.3 0.000000120031940.444444564476380.44444444444444 0.4 0.000000002082650.555555553472910.55555555555556 0.5 0.000000000182700.666666666849370.66666666666667 0.6 0.000004347869570.777773429908210.77777777777778 0.7 0.000000003546000.888888885342890.88888888888889 0.8 0.0000000000000011.00000000000000 0.9 -0.010111456888841.121222567999951.11111111111111 1.0   181 Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1918 Vol. 32 (1) 2019 Table 2. The performance of the train with epoch and time Table 3. Initial weight and bias of the network for different training algorithm Figure 1. Exact and Approximate solution of example1 using: Modify Trainlm & Trainlm Algorithms Example 2 In this example we will demonstrate the work of the offered algorithm on the first type of fractional equations for that integration [15] ∪ / 𝑑𝑡 35 24 0 1, where the exact solution is: ∪ 𝑡 MSE Time Epoch Performance of train Train Function 9.34772E‐06 0:00:02 50 2.39-31 Trainlm Weights and bias for trainlm Net.B{1}Net.LW{2,1}Net.IU{1,1}Net.IW{1,1} 0.48760.12390.98760.2436 0.25420.63980.98430.4563 0.19840.27600.09870.6534 0.12950.08340.23410.4237 0.59340.23140.75230.3122   182 Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1918 Vol. 32 (1) 2019 Table 4. Analytic and Neural solution of example 2 Table 5. The performance of the train with epoch and time MSE Time Epoch Performance of train Train Function 9.72845E-08 0:00:10 300 2.38-31 Trainlm Table 6. Initial weight and bias of the network for different training algorithm The error E(x)  | yt(x) ya(x) | w h e r e yt(x) computed by the following training algorithm( Trainlm) Out of suggested FFNN yt(x) for  LM training algorithm Analytic solution input Trainlmya(x) x 0.0000000000000000.000000000000000.0 0.000004654342980.099004654342980.099000000000000.1 0.000045366572450.192045366572450.192000000000000.2 0.000065872297870.273065872297870.273000000000000.3 0.000999435226040.335000564773960.336000000000000.4 0.000000564773420.375000564773420.375000000000000.5 0.000000354288780.384000354288780.384000000000000.6 0.000006755341980.357006755341980.357000000000000.7 0.000254463901340.287745536098660.288000000000000.8 0.000006453372430.171006453372430.171000000000000.9 0.0000000000000000.000000000000001.0 Weights and bias for trainlm Net.B{1} Net.LW{2,1} Net.IU{1,1} Net.IW{1,1} 0.12330.93650.09750.6574 0.75640.92840.43520.9786 0.87350.55420.75640.4352 0.92850.86350.97860.9807 0.56370.66570.98430.5543   183 Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1918 Vol. 32 (1) 2019 Figure 2. Exact & Approximate solution of example1 by using: Modify Trainlm & Trainlm Algorithms Example 3 In the third example we will demonstrate the proposed technique on the second type of fractional equations for that integration [16] 𝒖 𝟐 𝟏𝟔 𝟏𝟓 𝟓 𝟐 ∪ 𝒕 𝒕 𝟏 𝟐 𝒅𝒕 𝒙 𝟎 0 1, where the exact solution is ∪ 𝑡 Table 7. Analytic and Neural solution of example 3 The error E(x)  | yt(x) ya(x) | w h e r e yt(x) computed by the following training algorithm (Trainlm) Out of suggested FFNN yt(x) for LM training algorithm Analytic solution input Trainlmya(x) x 0.000000000000000.000000000000000.000000000000000.0 0.000004356208660.010004356208660.010000000000000.1 0.000000653231460.040000653231460.040000000000000.2 0.000000534274650.090000534274650.090000000000000.3 0.000000635277630.160000635277630.160000000000000.4 0.000000006242420.250000006242420.250000000000000.5 0.000003425534270.360003425534270.360000000000000.6 0.000342776134500.49034277613450.490000000000000.7 0.001436628759300.64143662875930.640000000000000.8 0.000342876143380.810342876143380.810000000000000.9 0.0000000000000011.000000000000001.0   184 Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1918 Vol. 32 (1) 2019 Table 8. The performance of the train with epoch and time MSE Time Epoch Performance of train Train Function 2.08999E-07 0:00:11 700 3.38-31 Trainlm Table 9. Initial weight and bias of the network for different training algorithm Figure 3. Exact & Approximate solution of example1 using: Modify Trainlm & Trainlm Algorithms 6. Conclusions In this paper, a new procedure was used to solve a special type of fractional integrals using parallel processors and a high-level training algorithm. Then applied in three examples of different types were the results compared with analytical solutions for these integrals. The results show the accuracy of suggested method is very high and the speed of convergence is ideal. Weights and bias for trainlm Net.B{1} Net.LW{2,1} Net.IU{1,1} Net.IW{1,1} 0.92750.95730.64770.7645 0.95390.36240.62980.8735 0.23650.87290.64290.1249 0.11260.75630.23140.9823 0.74030.83650.34250.8669   185 Ibn Al-Haitham Jour.for Pure&Appl.Sci. 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