Numerical approximation of fractional SEIR epidemic model of measles and smoking model by using Fibonacci wavelets operational matrix approach

In the present article, we have considered two essential models (The epidemic model of measles and the smoking model). Across the globe, the primary cause of health problems is smoking. Measles can be controlled in infectious populations using the mathematical model representing the direct transmission of infectious diseases. The Caputo fractional derivative operator of order α∈[0,1] is used to determine the solution of the system of fractional differential equations in both models. We employed the Fibonacci wavelets operational matrix of integration and developed the Fibonacci wavelet collocation method (FWCM) to solve a nonlinear coupled system of fractional differential equations (NCSFDEs). The proposed approach transforms the given model into a system of algebraic equations. The Newton-Raphson method is considered to extract the unknown coefficient values. Numerical outcomes are obtained to illustrate the simplicity and effectiveness of the proposed method. Graphs and tables show how consistently and effectively the developed strategy works. Mathematical software called Mathematica has been used to perform all calculations. Theorems explain the convergence of this approach.