Regions of existence for a class of nonlinear diffusion type problems

Amit Verma,Mandeep Singh,Ravi Agarwal

doi:10.2298/aadm190219013v

Abstract

The regions of existence are established for a class of two point nonlinear diffusion type boundary value problems (NDBVP) ?S??(x)- ns?(x)- m x s?(x) = f(x,s), m > 0, n ? R, x ? (0,1), s?(0) = 0, a1s(1) + a2s?(1) = C, where a1 > 0, a2 ? 0, C ? R. These problems arise very frequently in many branches of engineering, applied mathematics, astronomy, biological system and modern science (see the existing literature of this paper). By using the concept of upper and lower solutions with monotone constructive technique, we derive some sufficient conditions for existence in the regions where ?f/?s ? 0 and ?f/?s ? 0. Theoretical methods are applied for a set of problems which arise in real life.

Highlights

We study the class of two point nonlinear diffusion type boundary value problems (NDBVP), m
In this work, making use of constructive technique coupled with upper and lower solutions, existence theorems have been established for a form of two point NDBVP in the region
This section deals with some two point NDBVP that arise in various real life problems occurring in sciences and engineering

Summary

Introduction

We study the class of two point nonlinear diffusion type boundary value problems (NDBVP), m (1). Keywords and Phrases: Nonlinear diffusion boundary value problem,Monotone constructive technique, Upper and Lower solution, Green’s function, Real life application where a1 > 0, a2 ≥ 0, C ∈ R and the nonlinear function f (x, s) is continuous and Lipschitz in D := {(x, s) ∈ [0, 1] × R}. Zhang [30] observes and justifies that upper and lower solution technique is most promising technique to treat singular boundary value problems. In this work, making use of constructive technique coupled with upper and lower solutions, existence theorems have been established for a form of two point NDBVP in the region. The established theory is verified for some real life applications and summarized with concluding remark

Green’s function and its sign

Nonlinear Diffusion BVP : The Region of Existence

Real Life Applications

Conclusion