On a generalized fractional boundary value problem based on the thermostat model and its numerical solutions via Bernstein polynomials

Sina Etemad,Jehad Alzabut,Brahim Tellab,Chernet Tuge Deressa,Shahram Rezapour,Yongkun Li

doi:10.1186/s13662-021-03610-7

Abstract

In this paper, we introduce a new structure of the generalized multi-point thermostat control model motivated by its standard model. By presenting integral solution of this boundary problem, the existence property along with the uniqueness property are investigated by means of a special version of contractions named μ-φ-contractions and the Banach contraction principle. Then, on the given nonlinear generalized BVP of thermostat, the Bernstein polynomials are introduced and numerical solutions obtained by them are presented. At the end, three different structures of nonlinear thermostat models are designed and the results are examined.

Highlights

Fractional calculus and the existing notions in it are of high interest in different aspects of applied sciences, and one can find some instances of applications like signal and image processing, control theory, economics, optical systems, thermal materials, aerodynamics, mechanical systems, biology, and bio-mathematics [1,2,3,4,5,6,7,8,9,10]
Such a diversity and importance led to the publication of many research papers in this field, which revealed the flexibility of fractional calculus theory in designing various mathematical models
To follow the procedure of the paper, we introduce the operator K : X → X associated with the nonlinear thermostat generalized fractional boundary value problem (GFBVP) which takes the form (Ku)(s) =

Summary

Introduction

Fractional calculus and the existing notions in it are of high interest in different aspects of applied sciences, and one can find some instances of applications like signal and image processing, control theory, economics, optical systems, thermal materials, aerodynamics, mechanical systems, biology, and bio-mathematics [1,2,3,4,5,6,7,8,9,10]. Theorem 4.2 Let B(z) be the Bernstein vector introduced in (23) and I(ν) be the (m + 1) × (m + 1) operational matrix of the νth-FRL-integral which is formulated by. To determine an approximation of the exact solution u(s) by Bernstein polynomials, we utilize the FRL-integral of Bernstein polynomials with the operational matrix of the Caputo derivative used in [53]. It is easy to verify that our nonlinear thermostat GFBVP (40) satisfies all assumptions of Theorem 3.2, and its exact solution is u(s) = s2 + 2s3. Theorem 3.3 ensures the existence of a unique solution of the nonlinear thermostat GFBVP (42)

Conclusions

Methods