An efficient computational intelligence approach for solving fractional order Riccati equations using ANN and SQP

Abstract

a b s t r a c tA new computational intelligence technique is presented for solution of non-linear quadratic Riccati differential equations of fractional order based on artificial neural networks (ANNs) and sequential quadratic programming (SQP). The power of feed forward ANNs in an unsupervised manner is exploited for mathematical modeling of the equation; training of weights is carried out with an efficient constrained optimization technique based on the SQP algorithm. The proposed scheme is evaluated on two initial value problems of the Riccati fractional order equation with integer and non-integer derivatives. Comparison of results with the exact solution, and with reference numerical methods demonstrates the correctness of the proposed methodology. Performance of the proposed scheme is also validated using results of statistical analysis based on a sufficiently large number of independent runs.

Implicit hybrid methods for solving fractional Riccati equation

“…Since it is not easy task to find the exact solution of the fractional Riccati equation, several researchers investigated its solution numerically such as the Legendre wavelet operational matrix method [3], Adomian decomposition method [18], homotopy perturbation method [13], the Laplace transform and homotopy perturbation method [2], fractional Chebyshev finite difference method [10], the polynomial least squares method [4], and the Bezier curves [7]. In addition, artificial neural networks [20], the optimal homotopy asymptotic method [8] and the Laplace-Adomian-Pade method [12], Bäcklund transformation [17], and He's variational iteration method [9], are used to solved this problem. More methods can be found in [1,5,6,[22][23][24][25][26].…”

Dirichlet sınır değer koşullarına sahip adi diferansiyel denklemlerin nümerik çözümleri için basit tekrarlayan sinir ağları

“…The proposed approach is coded in Python and implemented on a Windows 64 bit operating system with a 3.4Ghz Intel(R) Core (TM) i7-2600 CPU and 16 Gb 800Mhz DDR3 RAM. (8) According to the mean of MSEs shown in Table 2, when ReLU are used in RNN as activation function, the best numerical solutions are obtained. Therefore, the absolute errors are listed in Table 3 with ReLU activation function.…”

Neural network methods to solve the Lane–Emden type equations arising in thermodynamic studies of the spherical gas cloud model

“…Few recent applications in this domain are stochastic numerical of nonlinear Jeffery-Hamel flow study in the presence of high magnetic field [34], problems arising in electromagnetic theory [35], modelling of electrical conducting solids [36], fuel ignition type model working in combustion theory [37], magnetohydrodynamics (MHD) studies [38], fluid mechanics problems [39], drainage problem [40], plasma physics problems [41], Bratu's problems [42], Van-der-Pol oscillatory problems [43], Troesch's problems [44], nanofluidic problems [45], multiwalled carbon nanotubes studies [46], nonlinear Painleve systems [47], nonlinear pantograph systems [48] and nonlinear singular systems [49][50][51][52][53]. Furthermore, the extended form of these methods has been applied to compute the solution of linear and nonlinear well-known fractional differential equations [54,55]. Keep viewing of these applications, authors are motivated to investigate in neural network methodologies to find the accurate and reliable solution for nonlinear singular systems based on LaneEmden type equations arising in thermodynamic studies of the spherical gas cloud model.…”