Abstract
This article developed a functional integration matrix via the Hermite wavelets and proposed a novel technique called the Hermite wavelet collocation method (HWM). Here, we studied two models: the coupled system of an ordinary differential equation (ODE) is modeled on the digestive system by considering different parameters such as sleep factor, tension, food rate, death rate, and medicine. Here, we discussed how these parameters influence the digestive system and showed them through figures and tables. Another fractional model is used on the COVID-19 pandemic. This model is defined by a system of fractional-ODEs including five variables, called S (susceptible), E (exposed), I (infected), Q (quarantined), and R (recovered). The proposed wavelet technique investigates these two models. Here, we express the modeled equation in terms of the Hermite wavelets along with the collocation scheme. Then, using the properties of wavelets, we convert the modeled equation into a system of algebraic equations. We use the Newton–Raphson method to solve these nonlinear algebraic equations. The obtained results are compared with numerical solutions and the Runge–Kutta method (R–K method), which is expressed through tables and graphs. The HWM computational time (consumes less time) is better than that of the R–K method.
Highlights
With the help of newly developed computational methods, many experts extracted more deep features of real world problems such as symmetry, optical waves, gravitational potentials and so on
We proposed a novel approach called Hermite wavelet collocation method (HWM) to solve the system of coupled ordinary differential equation (ODE)
This paper has successfully applied the HWM to the digestive model, which is assumed for sleep, tension, food, and death rates
Summary
With the help of newly developed computational methods, many experts extracted more deep features of real world problems such as symmetry, optical waves, gravitational potentials and so on. We proposed a novel approach called HWM to solve the system of coupled ODEs. The primary purpose is to present and explain a new numerical method for obtaining the approximate solution to the system of coupled equations of integer and fractional orders that cannot be solved exactly. Using the information with the vaccination, experts may observe more profound properties of the vaccination by using mathematical tools that are guaranteed scientifically This impels us to solve such equations via the Hermite wavelets method (HWM). The Hermite wavelet method is a new method, and here, we showed that it can be used to solve the biological models for both integer and fractional ordered coupled ODEs. The primary disadvantages of Runge–Kutta methods are that they require significantly more computer time than multi-step methods of comparable accuracy, and they do not yield good global estimates of the truncation error.
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