Somayeh EZADİ, Sahar AKSARI, Mitra JASEMI

Numerical Solution of ordinary Differential Equations Based on Semi-Taylor by Neural Network improvement

Abstract. In this paper, a new approach is proposed in order to solve the differential equations of ordinary initial value based on the feed-forward neural network and Semi-Taylor series ordinary differential equation is first replaced by a system of ordinary differential equations. A trial Solution of this System consists of two parts. The first part, that is, Semi-Taylor series contains no adjustable parameters. And the second part includes the neural network and adjustable parameters (the weights). Using modified neural network requires that training points be selected over the open interval (a, b) without training the network in the range of the first and end points. Therefore, the calculating volume containing computational error is reduced. In fact, the training points depending on the distance [a, b] selected for training neural networks are converted into similar points in the open interval (a, b) using the new approach, then the network is trained in these similar areas.In comparison with existing similar neural networks, the proposed model provides solutions with high accuracy. Numerical examples with simulation results illustrate the effectiveness of the proposed model.

Keywords:

Ordinary Differential Equations Semi-Taylor, MLP Neural Network, bfgs Technique,

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Table The optimal values of weights and biases. i 1 2 3 4 5 vi -0.7624 0038 -8458 5550 -0.6750 wi b 9275 2401 -5926 i -0138 0.0401 7595 EP(min) = 6.8078e-09
Table Comparison of the exact yand approximated y solutions. aand approximated t 6 5 4 3 2 1 i x 0.5 0.4 0.3 0.2 0.1 i 4256 2141 0151 0.8293 0.6574 0.5000 yt 4256 2141 0151 0.8293 0.6574 0.5000 ya 11 1 0.9 xt 0.8 0.7 0.6 6409 3802 1272 8831 6489 yt ya 6409 3802 1272 8831 6489
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