Fractal Fractional Order Operators in Computational Techniques for Mathematical Models in Epidemiology

New fractional operators, the COVID-19 model has been studied in this paper. By using different numericaltechniques and the time fractional parameters, the mechanical characteristics of the fractional order model areidentified. The uniqueness and existence have been established. Themodel’sUlam-Hyers stability analysis has beenfound. In order to justify the theoretical results, numerical simulations are carried out for the presented methodin the range of fractional order to show the implications of fractional and fractal orders.We applied very effectivenumerical techniques to obtain the solutions of themodel and simulations. Also, we present conditions of existencefor a solution to the proposed epidemicmodel and to calculate the reproduction number in certain state conditionsof the analyzed dynamic system. COVID-19 fractional order model for the case of Wuhan, China, is offered foranalysis with simulations in order to determine the possible efficacy of Coronavirus disease transmission in theCommunity. For this reason, we employed the COVID-19 fractal fractional derivative model in the example ofWuhan, China, with the given beginning conditions. In conclusion, again the mathematical models with fractionaloperators can facilitate the improvement of decision-making for measures to be taken in the management of anepidemic situation.

Chirped optical solitons and stability analysis of the nonlinear Schrödinger equation with nonlinear chromatic dispersion

The present paper aims to investigate the chirped optical soliton solutions of the nonlinear Schrödinger equation with nonlinear chromatic dispersion and quadratic-cubic law of refractive index.The exquisite balance between the chromatic dispersion and the nonlinearity associated with the refractive index of a fiber gives rise to optical solitons,which can travel down the fiber for intercontinental distances.The effective technique,namely,the new extended auxiliary equation method is implemented as a solution method.Different types of chirped soliton solutions including dark,bright,singular and periodic soliton solutions are extracted from the Jacobi elliptic function solutions when the modulus of the Jacobi elliptic function approaches to one or zero.These obtained chirped optical soliton solutions might play an important role in optical communication links and optical signal processing systems.The stability of the system is examined in the framework of modulational instability analysis.

New mathematical modelings of the human liver and hearing loss systems with fractional derivatives

Fractional derivatives are significant mathematical instruments that have been practiced to model real phenomena in different areas of science.Through the current investigation study,we develop the Human Liver and Hearing Loss models by employing three fractional operators called Atangana-Baleanu-Caputo,Caputo and Caputo-Fabrizio derivatives.The numerical techniques regarding the basic theories of fractional analysis considered to obtain the approximate solutions of the offered systems.Indeed,approximate results are presented for showing the efficacy of the methods during graphs of solutions.To see the performance of the suggested models,different fractional orders are tested.

Nonclassical Lie symmetry and conservation laws of the nonlinear timefractional Korteweg–de Vries equation

In this paper,we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann–Liouville concept.In this study,first,we employ the classical and nonclassical Lie symmetries(LS)to acquire similarity reductions of the nonlinear fractional far field Korteweg–de Vries(KdV)equation,and second,we find the related exact solutions for the derived generators.Finally,according to the LS generators acquired,we construct conservation laws for related classical and nonclassical vector fields of the fractional far field KdV equation.