Using collocation neural network to solve Eignvalue problems

MJ Journal on Applied Mathematics 1 (1) (2016) 1-8 Website: www.math-journals.com/index.php/JAM doi: 10.14419/jam.v1i1.21 Research paper

Using collocation neural network to solve Eignvalue problems Luma Naji Mohammed Tawfiq * Department of Mathematics, College of Education for pure science / Ibn Al-Haitham, Baghdad University *Corresponding author E-mail: dr.lumanaji@yahoo.com

Abstract The aim of this paper is to design collocation neural network to solve eignvalue problems for ordinary differential equations. The proposed network trained by back propagation with different training algorithms: quasi-Newton, Levenberg– Marquardt, and Bayesian Regulation. The next objective of this paper was to compare the performance of aforementioned algorithms with regard to predicting ability.

Keywords: Artificial Neural Network, Back Propagation, Training Algorithm, ODE, Eignvalue Problems.

AMS Subject Classification: 65Y05, 68M07, 68N30, 68T01, 68T05, 34L05, 34B15, 34L15, 34L1634L20

1. Introduction These days every process is automated. A lot of mathematical procedures have been automated. There is a strong need of software that solves differential equations (DEs) as many problems in science and engineering are reduced to differential equations through the process of mathematical modeling. Although model equations based on physical laws can be constructed, analytical tools are frequently inadequate for the purpose of obtaining their closed form solution and usually numerical methods must be resorted to.

Principal numerical methods available for solving DEs include the finite difference method (FDM), the finite element method (FEM), the finite volume method (FVM) and the boundary element method (BEM). All these methods generally require some discretization of the domain into a number of finite elements (Fes) which is not a straightforward task. In contrast to FE-type approximation, neural networks can be considered as approximation schemes where the input data for the design of a network consists of only a set of unstructured discrete data points. Thus an application of neural networks for solving DEs can be regarded as a mesh-free numerical method. It has been proved that feed forward neural networks FFNNs with one hidden layer are capable of universal approximation.

2. Related work Neural networks have found application in many disciplines: neurosciences, mathematics, statistics, physics, computer science and engineering. In the context of the numerical solution of differential equations, high-order derivatives are undesirable in general because they can introduce large approximation error. The use of higher order conventional Lagrange polynomials does not guarantee to yield a better quality (smoothness) of approximation. Many methods have been developed so far for solving differential equations, some of them produce a solution in the form of an array that contains the value of the solution at a selected group of points [1]. Others use basis functions to represent the solution in analytic form and transform the original problem usually to a system of algebraic equations [2]. Most of the previous study in solving differential equations using artificial neural network (Ann) is restricted to the case of solving the systems of algebraic equations which result from the discretisation of the domain [3]. Most of the previous works in solving differential equations using neural networks is restricted to the case of solving the linear systems of algebraic equations which result from the discretisation of the domain. The minimization of the networks energy function provides the solution to the system of equations [4]. Lagaris et al [5] employed two networks: a multilayer perceptron and a radial basis function network to solve partial differential equations (PDE) with boundary conditions (Dirichlet or Neumann) defined on boundaries with the case of complex boundCopyright Š 2016 Luma Naji Mohammed Tawfiq. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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ary geometry. Mc Fall et al [6] compared weight reuse for two existing methods of defining the network error function, weight reuse is shown to accelerate training of ODE, the second method outperforms the fails unpredictably when weight reuse is applied to accelerate solution of the diffusion equation. Tawfiq [7] proposed a radial basis function neural network (RBFNN) and Hopfield neural network (unsupervised training network ) as a designer network to solve ODE and PDE and compared between them. Malek et al [8] reported a novel hybrid method based on optimization techniques and neural networks methods for the solution of high order ODE which used three layered perceptron network. Akca et al [9] discussed different approaches of using wavelets in the solution of boundary value problems (BVP) for ODE, also introduced convenient wavelet representations for the derivatives for certain functions and discussed wavelet network algorithm. Mc Fall [10] presented multi-layer perceptron networks to solve BVP of PDE for arbitrary irregular domain where he used logsig. Transfer function in hidden layer and pureline in output layer and used gradient decent training algorithm, also, he used RBFNN for solving this problem and compared between them. Junaid et.al [11] used Ann with genetic training algorithm and log sigmoid function for solving first order ODE, Zahoor et al [12] has been used an evolutionary technique for the solution of nonlinear Riccati differential equations of fractional order and the learning of the unknown parameters in neural network has been achieved with hybrid intelligent algorithms mainly based on genetic algorithm (GA). Abdul Samath et al [13] suggested the solution of the matrix Riccati differential equation(MRDE) for nonlinear singular system using Ann. Ibraheem et al [14] proposed shooting neural networks algorithm for solving two point second order BVP in ODEs which reduced the equation to the system of two equations of first order. Hoda et al [4] described a numerical solution with neural networks for solving PDE, with mixed boundary conditions. Majidzadeh [15] suggested a new approach for reducing the inverse problem for a domain to an equivalent problem in a variational setting using radial basis functions neural network, also he used “cascade feed forward to solve two - dimensional Poisson equation with back propagation and levenberg Marqwardt train algorithm with the architecture three layers and 12 input nodes, 18 tansig. Transfer function in hidden layer, and 3 linear nodes in out put layer. Oraibi [16] designed feed forward neural networks (FFNN) for solving IVP of ODE. Ali [17] design fast FFNN to solve two point BVP. This paper proposed FFNN to solve two point singular boundary value problem (TPSBVP) with back propagation (BP) training algorithm. Tawfiq and Hussein [18] suggest multi-layer FFNN to solve singular boundary value problems.

3. What is artificial neural network?

Artificial neural network (Ann) is a simplified mathematical model of the human brain, It can be implemented by both electric elements and computer software. It is a parallel distributed processor with large numbers of connections, it is an information processing system that has certain performance characters in common with biological neural networks [19]. The arriving signals, called inputs, multiplied by the connection weights (adjusted) are first summed (combined) and then passed through a transfer function to produce the output for that neuron. The activation (transfer) function acts on the weighted sum of the neuron’s inputs and the most commonly used transfer function is the sigmoid function (tansig.) [17].

There are two main connection formulas (types): feedback (recurrent) and feed forward connection. Feedback is one type of connection where the output of one layer routes back to the input of a previous layer, or to same layer. Feed forward (FFNN) does not have a connection back from the output to the input neurons [20].

There are many different training algorithms but the most often used is the Delta-rule or back propagation (BP) rule. A neural network is trained to map a set of input data by iterative adjustment of the weights. Information from inputs is fed forward through the network to optimize the weights between neurons. Optimization of the weights is made by backward propagation of the error during training phase.

The Ann reads the input and output values in the training data set and changes the value of the weighted links to reduce the difference between the predicted and target (observed) values. The error in prediction is minimized across many training cycles (iteration or epoch) until network reaches specified level of accuracy. A complete round of forward–backward passes and weight adjustments using all input–output pairs in the data set is called an epoch or iteration. If a network is left to train for too long, however, it will be over trained and will lose the ability to generalize.

In this paper, we focused on the training situation known as supervised training, in which a set of input/output data patterns is available. Thus, the Ann has to be trained to produce the desired output according to the examples.

In order to perform a supervised training we need a way of evaluating the Ann output error between the actual and the expected output. A popular measure is the mean squared error (MSE) or root mean squared error (RMSE) [21].

4. Proposed design In the proposed approach the model function is expressed as the sum of two terms: the first term satisfies the boundary conditions (BC) and contains no adjustable parameters. The second term can be found by using Ann which is trained so as to satisfy the differential equation and such technique called collocation neural network.

In this section we will illustrate how our approach can be used to find the approximate solution of the general form 2nd order eignvalue problems (EVP): y"(x) = F( x, y(x), y'(x), λ ) Where a subject to certain BC’s and x  R, D  R denotes the domain and y(x) is the solution to be computed.

If yt (x, p) denotes a trial solution with adjustable parameters p, the problem is transformed to a discretize form:

(1)

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Min ďƒĽ G(x i , ď š t (x i , p), ďƒ‘ď š t (x i , p), ďƒ‘ 2ď š t (x i , p)) p

ˆ Minp x i ďƒŽD F (xi , yt(xi ,p), yt'(xi ,p), Îť )

(2)

Subject to the constraints imposed by the BC’s.

In the proposed approach, the trial solution yt employs a Ann and the parameters p correspond to the weights and biases of the neural architecture. We choose a form for the trial function yt(x) such that it satisfies the BC’s. This is achieved by writing it as a sum of two terms: yt (xi, p) = A(x) + G( x, N(x, p) )

(3)

Where N(x, p) is a single-output Ann with parameters p and n input units fed with the input vector x. The term A (x) contains no adjustable parameters and satisfies the BC’s. The second term G is constructed so as not to contribute to the BC’s, since yt(x) satisfy them. This term can be formed by using an Ann whose weights and biases are to be adjusted in order to deal with the minimization problem.

5. Computation of the gradient An efficient minimization of (2) can be considered as a procedure of training the Ann, where the error corresponding to each input xi is the value E (xi) which has to forced near zero. Computation of this error value involves not only the Ann output but also the derivatives of the output with respect to any of its inputs.

Therefore, in computing the gradient of the error with respect to the network weights consider a multi-layer Ann with n input units (where n is the dimensions of the domain) one hidden layer with H sigmoid units and a linear output unit .

For a given input x the output of the Ann is: n

H

ďƒĽ ď Ž ď ł(z )

N i 1

i

ďƒĽw x

i

, where zi 

j1

ij

j

 bi

wij denotes the weight connecting the input unit j to the hidden unit i vi denotes the weight connecting the hidden unit i to the output unit, bi denotes the bias of hidden unit i, and Ďƒ (z) is the sigmoid transfer function (tansig.).

The gradient of Ann, with respect to the parameters of the Ann can be easily obtained as: đ?œ•đ?‘ đ?œ•đ?‘Łđ?‘– đ?œ•đ?‘ đ?œ•đ?‘?đ?‘–

 ď ł (zi)

(4)

 viď łď‚˘(zi)

(5)

đ?œ•đ?‘ đ?œ•đ?‘Šđ?‘–đ?‘—

 viď łď‚˘(zi) xj

(6)

Once the derivative of the error with respect to the network parameters has been defined, then it is a straight forward to employ any minimization technique. It must also be noted, the batch mode of weight updates may be employed.

6. Illustration of the collocation neural network We will consider the 2nd order EVP: đ?œ•2 đ?‘Ś đ?œ•đ?‘Ľ 2

 f(x, y, y', Ν)

(7)

where x ďƒŽ [a, b] and the BC: y(a)  A, y(b) = B, ( Dirichlet case ) or y′(a)  A, y′(b) = B, ( Neumann case) or y′(a)  A, y (b) = B, (Mixed case). A trial solution can be written as: yt (x, p)  (bA– aB) / (b–a) + (B–A)x /(b–a) + (x–a)(x–b)N(x, p) Where N(x, p) is the output of a suggested network with one input unit for x and weights p.

Note that yt (x) satisfies the BC by construction. The error quantity to be minimized is given by:

(8)

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MJ Journal on Applied Mathematics  d t (x i )   f (x i ,  t (x i ))  i 1 d2dx yt (xi, p) / dx2 – f 2

n

E[p] 

 

(xi, yt(xi, p) , dyt(xi, p) / dx ) }2

(9)

Where the xi  [a, b]. Since: dyt (x, p) /dx  (B–A)/(b–a) + {(x–a) + (x–b)}N(x, p) + (x–a) (x–b)

dN(x, p) dx

And d2yt(x, p) /dx2 = 2N(x, p) + 2{(x–a) + (x–b)}

dN(x, p) dx

+ (x–a) (x–b) d2N(x, p) /dx2

It is straightforward to compute the gradient of the error with respect to the parameters p using (4) – (6). The same holds for all subsequent model problems.

7. Examples In this section we report numerical result, using a multi-layer FFNN having one hidden layer with 5 hidden units (neurons) and one linear output unit. The sigmoid activation of each hidden unit is tansig., the analytic solution y a(x) was known in advance. Therefore we test the accuracy of the obtained solutions by computing the deviation: y(x)  | yt(x) – ya(x) |.

In order to illustrate the characteristics of the solutions provided by the neural network method, we provide figures displaying the corresponding deviation y(x) both at the few points (training points) that were used for training and at many other points (test points) of the domain of equation. The latter kind of figures are of major importance since they show the interpolation capabilities of the neural solution which to be superior compared to other solution obtained by using other methods. Moreover, we can consider points outside the training interval in order to obtain an estimate of the extrapolation performance of the obtained numerical solution.

Example 1 Consider the following 2nd order EVP: y ′′ − λ y′ + y = 0 With Dirishlit BC: y(0) 1 , y(1) = e1 , and x [0, 1].

The analytic solution is: ya (x)  ex , according to (8) the trial neural form of the solution is taken to be: yt (x) 1 + (e1 − 1)x + x(x − 1)N(x, p), and λ= -2 The Ann trained using a grid of ten equidistant points in [0, 1]. Figure (1) display the analytic and neural solutions with different training algorithm. The neural results with different types of training algorithm such as: Levenberg – Marquardt ( trainlm ), quasi – Newton ( trainbfg ), Bayesian Regulation ( trainbr ) introduced in Table (1) and its errors given in Table (2), Table (3) gives the performance of the train with epoch and time and Table(4) gives the weight and bias of the designer network 2.8 ya lm bfg br

2.6 2.4 2.2

yt

2 1.8 1.6 1.4 1.2 1

0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

Fig. 1: Analytic and Neural Solution of Example 1 Using Different Training Algorithm.

Table 1: Analytic and Neural Solution of Example 1 Input X 0.0 0.1 0.2 0.3 0.4

Analytic Solution Ya(X) 1 1.105170918075648 1.2214027576003 1.349858807576003 1.491824697641270

Output Of Suggested FFNN Yt(X) For Different Training Algorithm Trainlm Trainbfg Trainbr 1.000006207608819 1.000003510742645 1.000000002642383 1.105170918075648 1.105170920551764 1.105170903369380 1.221402758160170 1.221402764357907 1.221402791470623 1.349859412929561 1.349858811210834 1.349858772797468 1.491824697641270 1.491824701249854 1.491824700915662

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0.5 0.6 0.7 0.8 0.9 1.0

1.648721270700128 1.822118800390509 2.013752707470477 2.225540928492468 2.459603111156950 2.718281828459046

5

1.648721270700127 1.822120455330252 2.013755100215994 2.225540928492467 2.459603111156950 2.718281828459045

1.648721272813596 1.822118474463477 2.013752201382579 2.225540932561377 2.459603109698643 2.718281832254779

1.648721308524495 1.822118747621918 2.013752745457101 2.225540912067933 2.459603115260220 2.718281827995976

Table 2: Accuracy of Solutions for Example 1 The error E(x)  | yt(x) ya(x) | w h e r e yt(x) computed by the following training algorithm Trainlm Trainbfg Trainbr 6.207608818975530e-06 3.510742645040921e-06 2.642383423179240e-09 2.220446049250313e-16 2.476116200966771e-09 1.470626820321286e-08 2.220446049250313e-16 6.197737434376904e-09 3.331045328458515e-08 6.053535575034630e-07 3.634830880727691e-09 3.477853538491615e-08 4.440892098500626e-16 3.608583432068713e-09 3.274391868757220e-09 8.881784197001252e-16 2.113467623132692e-09 3.782436719390603e-08 1.654939743023220e-06 3.259270320299379e-07 5.276859105229903e-08 2.392745516832662e-06 5.060878973672800e-07 3.798662406850895e-08 4.440892098500626e-16 4.068909209564708e-09 1.642453506534025e-08 0 1.458307252732993e-09 4.103269723998437e-09 4.440892098500626e-16 3.795733505285170e-09 4.630695826790543e-10 Table 3: The Performance of the Train for Epoch and Time Train Function Trainlm

Performance of train 4.93e-32

Epoch 515

Time 0:00:03

msereg 3.8753e-012

Trainbfg

2.02e-24

438

0:00:04

1.0381e-012

Trainbr

8.50e-15

20468

0:02:44

6.9530e-016

Table 4: Initial Weight and Bias of the Network for Different Training Algorithm

Net.IW{1,1} 0.9969 0.5535 0.5155 0.3307 0.4300 Net.IW{1,1} 0.5502 0.6448 0.8451 0.5950 0.3310 Net.IW{1,1} 0.2133 0.3829 0.0297 0.4723 0.3334

Weights and bias for trainlm Net.LW{2,1} 0.4918 0.0710 0.8877 0.0646 0.4362 Weights and bias for trainbfg Net.LW{2,1} 0.2114 0.6257 0.2631 0.9756 0.1614 Weights and bias for trainbr Net.LW{2,1} 0.9758 0.5554 0.8463 0.4081 0.4620

Example 2: Consider the following 2nd order EVP: y′′ − 6y′ + (9 + λ)y = 0 With Dirishlit BC: y (0)  0, y (1) = 0, and x [0, 1]. The analytic solution is: The analytic solution is: ya (x)  e3x sin(πx) According to (8) the trial neural form of the solution is taken to be: yt (x) x(x − 1)N(x, p) and λ = 9.8778204

Net.B{1} 0.8266 0.3945 0.6135 0.8186 0.8862 Net.B{1} 0.7945 0.9244 0.4663 0.6136 0.4105 Net.B{1} 0.8263 0.9912 0.5239 0.9254 0.7390

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Neural solutions with different training algorithm. The neural results with different types of training algorithm such as: Levenberg – Marquardt (trainlm), quasi – Newton (trainbfg), Bayesian Regulation (trainbr) introduced in Table (5) and its errors given in Table (6), Table (7) gives the performance of the train with epoch and time and Table (8) gives the weight and bias of the designer network.

7 ya lm bfg br

6 5

yt

4 3 2 1 0 -1

0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

Fig. 2: Analytic and Neural Solution of Example 2 Using Different Training Algorithm.

Table 5: Analytic and Neural Solution of Example 2 Input x 0 0.0909 0.1818 0.2727 0.3636 0.4545 0.5455 0.6364 0.7273 0.8182 0.9091 1.0000

Analytic solution ya(x) 0 0.370067534135651 0.932817597286683 1.712812248886422 2.707955518449923 3.870584966600148 5.084175752745671 6.137243789258745 6.697758869064653 6.293675942244905 4.308008314201565 2.459768850213775e-15

Output of suggested FFNN yt(x) for different training algorithm Trainlm Trainbfg Trainbr -2.974402443929171e-15 -0.001601878820345 -2.605985521217994e-08 0.369220195791156 0.369431779689253 0.370068116352455 0.932817597286683 0.932817597143824 0.932814218038194 1.712812248886423 1.712812248656277 1.712821976905245 2.707595635438058 2.707955518552264 2.707938727680147 3.870584966600146 3.870584966556053 3.870603938718505 5.084175752745672 5.083836984900312 5.084161044545271 6.133898832162847 6.137243789270370 6.137251818008972 6.690006860685525 6.697758869170623 6.697755736272486 6.293675942244903 6.284032374483856 6.293676822256024 4.308008314201563 4.308008314231338 4.308008139655482 7.436006109822926e-16 -5.980184833641793e-11 2.078956059942216e-08 Table 6: Accuracy of Solutions for Example 2

The error E(x)  | yt(x) ya(x) | w h e r e yt(x) computed by the following training algorithm Trainlm Trainbfg Trainbr 2.974402443929171e-15 0.001601878820345 2.605985521217994e-08 8.473383444949634e-04 6.357544463979070e-04 5.822168037683539e-07 4.440892098500626e-16 1.428595020058765e-10 3.379248489698838e-06 4.440892098500626e-16 2.301454582465112e-10 9.728018822485751e-06 3.598830118649588e-04 1.023403584099469e-10 1.679076977589844e-05 2.220446049250313e-15 4.409494991364227e-11 1.897211835633073e-05 8.881784197001252e-16 3.387678453590226e-04 1.470820040072596e-05 0.003344957095899 1.162447915703524e-11 8.028750226429793e-06 0.007752008379128 1.059703436112613e-10 3.132792167015452e-06 1.776356839400251e-15 0.009643567761048 8.800111190865323e-07 2.664535259100376e-15 2.977262880676790e-11 1.745460833646462e-07 1.716168239231483e-15 5.980430810526815e-11 2.078955813965331e-08 Table 7: The Performance of the Train with Epoch and Time Train Function Trainlm

Performance of train 1.41e-30

Epoch 142

Time 0:00:01

Msereg.

5.4097e-006

Trainbfg

2.98e-22

1116

0:00:12

7.2063e-006

Trainbr

1.04e-09

3602

0:00:28

7.7976e-011

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Table 8: Weight and Bias of the Network for Different Training Algorithm Weights and bias for trainbfg Net.IW{1,1} Net.LW{2,1} 0.3065 0.5811 0.7315 0.8083 0.4599 0.0504 0.4050 0.1487 0.2517 0.4914 Weights and bias for trainlm Net.IW{1,1} Net.LW{2,1} 0.7452 0.8287 0.3371 0.6859 0.5843 0.2673 0.4690 0.9695 0.0873 0.1838

Net.IW{1,1} 0.4402 0.1568 0.3260 0.3141 0.8945

8.

Weights and bias for trainbr Net.LW{2,1} 0.2470 0.3107 0.4089 0.7080 0.1436

Net.B{1} 0.8917 0.3130 0.2828 0.0647 0.0605 Net.B{1} 0.2999 0.4112 0.2365 0.1951 0.7054

Net.B{1} 0.8713 0.0832 0.4617 0.0304 0.7532

Conclusion

The paper explains design and development of an application for solving eign-value problem of ordinary differential equations. Graphs are output when the user specifies input in the form of a equation. Often it is not possible for everyone to know to solve differential equations. People who are proficient in differential equation may find it time consuming to keep on solving simple equations which they can use as intermediate results. The paper aims at providing such users with a tool to develop fast solutions. The output is not just the graph but the list of values for x through the solution space.

References [1] Mohammed, K. M., (2009), On Solution of Two Point Second Order Boundary Value Problems by Using Semi-Analytic Method, MSc thesis, University of Baghdad, College of Education – Ibn – Al- Haitham.

[2] LeVeque, R. J., (2006), Finite Difference Methods for Differential Equations, University of Wash-ington, AMath 585, Winter Quarter.

[3] Agatonovic-Kustrin, S. and Beresford, R., (2000), Basic concepts of artificial neural network modeling and its application in pharmaceutical, research. J. Pharm. Biomed. Anal., Vol. 22, pp: 717–727. http://dx.doi.org/10.1016/S0731-7085(99)00272-1.

[4] Hoda, S. A., and Nagla, H. A., (2011), On Neural Network Methods for Mixed Boundary Value Problems, International Journal of Nonlinear Science, Vol.11, No.3, pp: 312-316.

[5] Lagaris, I. E., Likas, A. C., and Papageorgiou, D. G., September (2000), Neural-Network Methods for Boundary Value Problems with Irregular Boundaries, IEEE Transactions on Neural Networks, Vol. 11, No. 5, pp: 1041- 1049.

[6] Mc Fall, K. S., and Mahan, J. R., (2004), Investigation of Weight Reuse in Multi-Layer Perceptron Networks for Accelerating the Solution of Differential Equations, IEEE International Conference on Neural Networks, Vol.14, pp: 109 -114.

[7] Tawfiq, L. N. M., (2004), Design and Training Artificial Neural Networks for Solving Differential Equations, PhD thesis, University of Baghdad , College of Education – Ibn – Al- Haitham.

[8] Malek, A., and Beidokhti, R. S., (2006), Numerical solution for high order differential equations using a hybrid neural networkoptimization method, Applied Mathematics and Computation ,Vol.183, pp: 260–271. http://dx.doi.org/10.1016/j.amc.2006.05.068.

[9] Akca, H., Al-Lail, M. H. and Covachev, V., (2006), Survey on Wavelet Transform and Application in ODE and Wavelet Networks, Advances in Dynamical Systems and Applications, Vol. 1, No. 2, pp. 129 –162.

[10] Mc Fall, K. S., (2006), An Artificial Neural Network Method for Solving Boundary Value Prob-lems with Arbitrary Irregular Boundaries, PhD Thesis, Georgia Institute of Technology.

[11] Junaid, A., Raja, M. A. Z., and Qureshi, I. M., (2009), Evolutionary Computing Approach for The Solution of Initial value Problems in Ordinary Differential Equations, World Academy of Science, En-gineering and Technology, No.55 , PP:578 – 581.

[12] Zahoor, R. M. A., Khan, J. A., and Qureshi, I. M., (2009), Evolutionary Computation Technique for Solving Riccati Differential Equation of Arbitrary Order, World Academy of Science, Engineering and Technology, Vol. 58 , pp: 303 – 309.

[13] Abdul Samath, J., Kumar, P. S. and Begum, A., (2010), Solution of Linear Electrical Circuit Prob-lem using Neural Networks, International Journal of Computer Applications, Vol. 2, No.1, pp: 6-13.

[14] Ibraheem, K. I., and Khalaf, B. M., June (2011), Shooting Neural Networks Algorithm for Solv-ing Boundary Value Problems in ODEs, Applications and Applied Mathematics: An International Journal, Vol. 6, Issue 11, pp. 1927 – 1941.

[15] Majidzadeh, K., (2011), Inverse Problem with Respect to Domain and Artificial Neural Network Algorithm for the Solution, Mathematical Problems in Engineering, Vol. 2011, pp: 1-16. http://dx.doi.org/10.1155/2011/145608.

[16] Oraibi, Y. A., (2011), Design Feed Forward Neural Networks For Solving Ordinary Initial Value Problem, MSc Thesis, University of Baghdad, College of Education Ibn Al-Haitham.

[17] Ali, M. H., (2012), Design Fast Feed Forward Neural Networks to Solve Two Point Boundary Value Problems, MSc Thesis, University of Baghdad, College of Education Ibn Al-Haitham.

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[18] Tawfiq, L. N. M., and Hussein, A. A. T., (2013), Design Feed Forward Neural Network to Solve Singular Boundary Value Problems, ISRN Applied Mathematics, Vol. 2013, Article ID 650467, pp: 1-7.

[19] Galushkin, I. A., (2007), Neural Networks Theory, Berlin Heidelberg.

[20] Mehrotra, K., Mohan, C. K., and Ranka, S., (1996), Elements of Artificial Neural Networks, Springer.

[21] Ghaffari, A., Abdollahi, H., Khoshayand, M. R., Soltani Bozchalooi, I., Dadgar, A., Rafiee-Tehrani, M., (2006), Performance comparison of neural network training algorithms in modeling of bimodal drug delivery, International Journal of Pharmaceutics, Vol. 327, pp: 126 – 138. http://dx.doi.org/10.1016/j.ijpharm.2006.07.056.