Caputo fractional reduced differential transform method for SEIR epidemic model with fractional order

This paper proposes the Caputo Fractional Reduced Differential Transform Method (CFRDTM) for Susceptible-Exposed-Infected-Recovered (SEIR) epidemic model with fractional order in a host community. CFRDTM is the combination of the Caputo Fractional Derivative (CFD) and the well-known Reduced Differential Transform Method (RDTM). CFRDTM demonstrates feasible progress and efficiency of operation. The properties of the model were analyzed and investigated. The fractional SEIR epidemic model has been solved via CFRDTM successfully. Hence, CFRDTM provides the solutions of the model in the form of a convergent power series with easily computable components without any restrictive assumptions.

Construction of two-dimensional flows in physical space arising after the decay of a special discontinuity

Рассмотрены двумерные изэнтропические течения политропного газа, возникающие в начальный момент времени после мгновенного разрушения непроницаемой стенки, отделяющей неоднородный покоящийся газ от вакуума. В качестве математической модели используется система уравнений газовой динамики с учетом действия силы тяжести. В системе уравнений газовой динамики вводится автомодельная особенность в переменную x и для полученной системы ставится задача Коши с данными на звуковой характеристике. Решение начально-краевой задачи строится в виде степенного ряда. Коэффициенты ряда находятся при интегрировании обыкновенных дифференциальных уравнений. Для доказательства сходимости этого ряда ставится начально-краевая задача в пространстве других независимых переменных, а решение строится в виде своего сходящегося степенного ряда, и доказывается эквивалентность решений первой и второй начально-краевых задач The aim of this study is to construct a solution to the problem of the decay of a special discontinuity in physical space, i.e., two-dimensional isentropic flows of polytropic gas, arising after the instantaneous destruction of an impenetrable wall that separates an inhomogeneous resting gas from a vacuum. The study takes into account the effect of gravity. Research Methods. A variable, which governs the evolution of the self-similar singularity from the initial interface is introduced into the system of equations of gas dynamics. For the resulting system, the Cauchy problem is posed with prescribed values on the sound characteristic. The solution to this problem is constructed in the form of power series. The coefficients of the series are determined by solving algebraic and ordinary differential equations. Further, to prove the convergence of this series, an initial-boundary-value problem is posed in a special functional space. The solution to this initial-boundary value problem is constructed in the form of its convergent power series and the equivalence of solutions for the first and second initial-boundary value problems is proved. Solutions of the problem for the decay of a special discontinuity are constructed in the form of convergent power series. The equivalence of solutions in the physical and special functional space is proved. Conclusions. The solution constructed in physical space determines the initial conditions for the difference scheme for the numerical simulation of the given characteristic Cauchy problem, while the one, built in a special functional space, allows setting boundary conditions for this scheme

Homotopy Transforms Analysis Method for Solving Fractional Navier- Stokes Equations with Applications

The presented work includes the Homotopy Transforms of Analysis Method (HTAM). By this method, the approximate solution of nonlinear Navier- Stokes equations of fractional order derivative was obtained. The Caputo's derivative was used in the proposed method. The desired solution was calculated by using the convergent power series to the components. The obtained results are demonstrated by comparison with the results of Adomain decomposition method, Homotopy Analysis method and exact solution, as explained in examples (4.1) and (4.2). The comparison shows that the used method is powerful and efficient.