Forecasting of fast fluctuated and high-frequency financial data is always a challenging problem in the field of economics and modelling. In this study, a novel hybrid model with the strength of fractional order derivative is presented with their dynamical features of deep learning, long-short term memory (LSTM) networks, to predict the abrupt stochastic variation of the financial market. Stock market prices are dynamic, highly sensitive, nonlinear and chaotic. There are different techniques for forecast prices in the time-variant domain and due to variability and uncertain behavior in stock prices, traditional methods, such as data mining, statistical approaches, and non-deep neural networks models are not suited for prediction and generalized forecasting stock prices. While autoregressive fractional integrated moving average (ARFIMA) model provides a flexible tool for classes of long-memory models. The advancement of machine learning-based deep non-linear modelling confirms that the hybrid model efficiently extracts profound features and model non-linear functions. LSTM networks are a special kind of recurrent neural network (RNN) that map sequences of input observations to output observations with capabilities of long-term dependencies. A novel ARFIMA-LSTM hybrid recurrent network is presented in which ARFIMA model-based filters having the linear tendencies better than ARIMA model in the data and passes the residual to the LSTM model that captures nonlinearity in the residual values with the help of exogenous dependent variables. The model not only minimizes the volatility problem but also overcome the over fitting problem of neural networks. The model is evaluated using PSX company data of the stock market based on RMSE, MSE and MAPE along with a comparison of ARIMA, LSTM model and generalized regression radial basis neural network (GRNN) ensemble method independently. The forecasting performance indicates the effectiveness of the proposed AFRIMA-LSTM hybrid model to improve around 80% accuracy on RMSE as compared to traditional forecasting counterparts.

This research investigates the heat and mass transfer in 3-D MHD radiative flow of water based hybrid nanofluid over an extending sheet by employing the strength of numerical computing based Lobatto IIIA method. Nanoparticles of aluminum oxide (Al₂O₃) and silver (Ag) are being used with water (H₂O) as base fluid. By considering the heat transfer phenomenon due to thermal radiation effects. The physical flow problem is then modeled into set of PDEs, which are then transmuted into equivalent set of nonlinear ODEs by utilizing the appropriate similarity transformations. The system of ODEs is solved by the computational strength of Lobatto IIIA method to get the various graphical and numerical results for analyzing the impact of various physical constraints on velocity and thermal profiles. Additionally, the heat transfers and skin friction analysis for the fluid flow dynamics is also investigated. The relative errors up to the accuracy level of 1e-15, established the worth and reliability of the computational technique. It is observed that heat transfer rate increases with the increase in magnetic effect, Biot number and rotation parameter.

The technique of Levenberg–Marquardt back propagation with neural networks (TLMB-NN) was used in this research article to investigate the heat transfer of Maxwell base fluid flow of nanomaterials (HTM-BFN) with MHD over vertical moving surfaces. In this study, the effects of thermal energy, concentration, and Brownian motion are also employed. Moreover, the impacts of a heat-absorbing fluid with viscous dissipation and radiation have been explored. To simplify the governing equations from a stiff to a simple system of non-linear ODEs, we exploited the efficacy of suitable similarity transformation mechanism. Through applicability of state-of-the-art Adams numerical technique, a set of data for suggested (TLMB-NN) is generated for several situations (scenarios) by changing parameters, such as the Thermophoresis factor N_t, Hartmann number M, Eckert number E_c, concentration Grashoff parameter G_c, Prandtl number P_r, Lewis number Le, thermal Grashof number G_T, and Brownian motion factor N_b. The estimate solution of different instances has validated using the (TLMB-NN) training, testing, and validation method, and the recommended model was compared for excellence. Following that, regression analysis, mean square error, and histogram explorations are used to validate the suggested (TLMB-NN). The proposed technique is distinguished based on the proximity of the proposed and reference findings, with an accuracy level ranging from 10⁻⁹ to 10⁻¹⁰.