In the present paper, we employ a wavelets optimization method is employed for the elucidations of fractional partial differential equations of pricing European option accompanied by a Lévy model. We apply the Legendre wavelets optimization method (LWOM) to optimize the governing problem. The novelty of the proposed method is the inclusion of differential evolution algorithm (DE) in the Legendre wavelets method for the optimized approximations of the unknown terms of the Legendre wavelets. Sequentially, the functions and components of the pricing models are discretized by utilizing the operational matrix of fractional integration of Legendre wavelets. Illustratively, the implementation of the LWOM is exemplified on a pricing European option Lévy model and successfully depicted the stock paths. Moreover, comparison analysis of the Black-Scholes model with a class of Lévy model and LWOM with q-homotopy analysis transform method (q-HATM) is also deliberated out. Accordingly, the technique is found to be appropriate for financial models that can be expressed as partial differential equations of integer and fractional orders, subjected to initial or boundary conditions.

In this paper, a process is devised systematically to scrutinize the scrolling chaotic behaviour of fractional-order Chua's system. The process is composed of fractional Laplace transformation, artificial neural network with Mexican hat wavelet as an activation function and simulated annealing. Sequentially, the parametric expansion of fractional Laplace transform is employed to convert the governing fractional system into an ordinary differential system. Next, artificial neural network and simulated annealing approximate and optimize the attained system and produce accurate solutions. The predictability and elaboration of double scrolling chaotic structures of fractional-order Chua's system are also studied using the Lyapunov exponent and fifth-fourth Runge-Kutta method. Moreover, the mean absolute error and root mean square error are measured for the convergence analysis of the proposed scheme. On the whole, the accurate approximate solutions, the phase plots of Lyapunov exponent spectrum and bifurcation maps of the dynamical evolution of fractional Chua's system are a triumph of this endeavour.

Microalgae and cyanobacteria have sparked a lot of interest due to their potential in various industries like biorefineries, biopharmaceuticals, food supplements, nutraceuticals, and other high‐value products. Polysaccharides, vitamins, proteins, enzymes, and steroids are valuable products isolated from microalgae and cyanobacteria and potentially used in health and biomedical applications. Bioactive compounds derived from microalgae and cyanobacteria exhibit various pharmaceutical properties like antibacterial, anticancer, antiviral, antialgal, and antioxidant. From the properties listed above, the research for novel antibiotics has become particularly appropriate. In addition, the possible emergence of resistance against pathogens, as well as the potential decline in antibiotic efficacy, has prompted researchers to look for a new source of antibiotics. Microalgae and cyanobacteria have indicated a great and unexplored potential among these sources. For this reason, microalgae and cyanobacteria have been highlighted for their efficiency in different industrial sectors, as well as for their potential uses in the betterment of human and environmental health. This review gives an overview of bioactive compounds and metabolites with several biological properties isolated from microalgae and cyanobacteria for treating different animal and human diseases.

The purpose of this paper is to investigate the chaotic influence of the fractional order jerk system with the theoretical execution of circuit and practical consequent on cryptography. The Caputo fractional derivative (CFD) has been used for the commensurate order and obtains the necessary condition to appear chaos using Lyapunov exponent (LE). The existence, uniqueness of the system is analyzed, and the stabilities of the equilibrium points are explored. The output of the electronic circuit equations has been inspected graphically and the values of passive components resistor, capacitor and voltage are tabulated. The scrambling protocol is designed in multiple paradigmatic language Python for various aspects of information security such as sound and image for confidentiality. The analog and numerical simulations carry out in Multisim and Mathematica respectively, to see the effects of physical parameters on phase portraits which are incorporated through graphs and tables. The significance level of occur fluctuation is also measured by a statistical package the NIST test suite to ensure the stream of generating numbers for a particular cryptographic application. Furthermore, originality of the system also tested with the help of error measuring tool named mean absolute error (MAE) and found that the performance index of the designed system is in good agreement.

This paper presents an approximate solution of nonlinear fractional differential equations (FDEs) that exhibit an oscillatory behavior by using a metaheuristic technique. The solutions of the governing equations are approximated by using homotopy perturbation method (HPM) along with the fractional derivative in the Caputo sense. The designed methodology is based on a weighted series of HPM in conjunction with a nature-inspired algorithm. The idea is instantly fascinated by the researchers on the consequent implementation of nature-inspired learning algorithms such as a Cuckoo search algorithm (CSA). The usage of CSA has accelerated the minimized search path of error to the convergent values of the solution. The validity and accuracy of the proposed technique are ascertained by calculating the approximate solution and the error norms which ensure the convergence of the approximation that can be further increased. The critical analysis is also provided by the numerical simulation of two different test models. Discussion of key points has been determined by the tabulation of numerical values and graphs. Comparative study of the results with known numerical technique is also performed.

The convection differential models play an essential role in studying different chemical process and effects of the diffusion process. This paper intends to provide optimized numerical results of such equations based on the conformable fractional derivative. Subsequently, a well-known heuristic optimization technique, differential evolution algorithm, is worked out together with the Taylor's series expansion, to attain the optimized results. In the scheme of the Taylor optimization method (TOM), after expanding the functions with the Taylor's series, the unknown terms of the series are then globally optimized using differential evolution. Moreover, to illustrate the applicability of TOM, some examples of linear and non-linear fractional convection diffusion equations are exemplified graphically. The obtained assessments and comparative demonstrations divulged the rapid convergence of the estimated solutions towards the exact solutions. Comprising with an effective expander and efficient optimizer, TOM reveals to be an appropriate approach to solve different fractional differential equations modeling various problems of engineering.

In this paper, we propose an innovative chaotic system, combining fractional derivative and sine-hyperbolic nonlinearity with circuit execution. The study of this system is conducted via an analog circuit simulator, using two anti-parallel semiconductor diodes to provide hyperbolic sine nonlinearity, and to function as operational amplifiers. The multi-stability of the system is also enhanced with the help of multi-equilibrium points for distinct real orders of system. The system explores the generation of a four-wing attractor in different phases, both numerically and electronically. By changing the input parameters of the system, different graphs are generated for current flow in state, phase, and space, to confirm the precision of the fractional order derivatives. A reasonable simulation shows that the deliberate circuit provides effective chaos in terms of speed and accuracy, which is comensurate with the numerical simulation. This nonlinear chaotic behavior is utilized to encrypt sound (.wav), images (.jpg), and animated (.gif) data which are a major requirement for the security of communication systems. The proposed circuit performs chaos and cryptographic tasks with high-effective analog computation, and constitutes a novel approach to this area of research.

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