A new stochastic approach for solution of Riccati differential equation of fractional order

Abstract

In this article, a stochastic technique has been developed for the solution of nonlinear Riccati differential equation of fractional order. Feed-forward artificial neural network is employed for accurate mathematical modeling and learning of its weights is made with heuristic computational algorithm based on swarm intelligence. In this scheme, particle swarm optimization is used as a tool for the rapid global search method, and simulating annealing for efficient local search. The scheme is equally capable of solving the integer order or fractional order Riccati differential equations. Comparison of results was made with standard approximate analytic, as well as, stochastic numerical solvers and exact solutions.

Legendre Wavelet Operational Matrix Method for Solution of Riccati Differential Equation

“…It can be seen from Figure 1 that the solution obtained by the proposed LWM approach is more close to the exact solution. Table 1 describes the efficiency of the proposed method by comparing with the methods in [20,22] through their absolute error. The following is used for the errors of the approximation̂( ) of ( ); that is, ‖ −̂‖ = max | ( )−̂( )|.…”

“…Results obtained by LWM for = 1, = 2, and = 3 are shown in Figure 3 and it can be seen from the figure that solution given by the LWM merely coincides with the exact solution. Figure 4 shows the obtained results of (62) and (63) Table 2 describes the efficiency of the proposed method by comparing with the methods in [20,22] through their absolute error. Table 1 shows that very high accuracies are obtained for = 3 and = 5 by the present method and from these results we can identify that guarantee of convergence of the proposed LWM approach is very high.…”

“…Raja et al [20] developed a stochastic technique based on particle swarm optimization and simulated annealing. They were used as a tool for rapid global search method and simulated annealing for efficient local search method.…”

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.…”

Comparison of three unsupervised neural network models for first Painlevé Transcendent

“…Moreover, new applications of these schemes are reported to solve fractional differential equations as well [34][35][36]. For example, Riccati and Bagley-Torvik equations have also been solved effectively by the said procedure [37,38]. Stochastic solvers based on evolutionary computing [39] and swarm intelligence [40] have been applied to solve the Painlevé equation I, which provide reliable and accurate solution of the problem.…”