Regions of existence for a class of nonlinear diffusion type problems

TL;DR

This paper establishes the existence regions for solutions of a class of nonlinear diffusion boundary value problems using upper and lower solutions, applicable in various scientific fields.

Contribution

It introduces sufficient conditions for solution existence based on the monotonicity of the nonlinear term, expanding the theoretical framework for nonlinear diffusion problems.

Findings

Existence regions depend on the sign of the partial derivative of f with respect to s.

Sufficient conditions are derived for both non-negative and non-positive derivatives.

Applicable to real-world problems in engineering, biology, and physics.

Abstract

The regions of existence are established for a class of two point nonlinear diffusion type boundary value problems (NDBVP) \begin{eqnarray*} &&\label{abst-intr-1} -s''(x)-ns'(x)-\frac{m}{x}s'(x)=f(x,s), \qquad m>0,~n\in \mathbb{R},\qquad x\in(0,1),\\ &&\label{abst-intr-2} s'(0)=0, \qquad a_{1}s(1)+a_{2}s'(1)=C, \end{eqnarray*} where $a_{1}>0,$ $a_{2}\geq0,~ C\in\mathbb{R}$. These problems arise very frequently in many branches of engineering, applied mathematics, astronomy, biological system and modern science (see \cite{Gatica1989, GRAY1980, Baxley1991, Chandershekhar1939, Duggan1986, Chambre1952}). By using the concept of upper and lower solutions with monotone constructive technique, we derive some sufficient conditions for existence in the regions where $\frac{\partial f}{\partial s}\geq0$ and $\frac{\partial f}{\partial s}\leq0$. Theoretical methods are applied for a set of problems which arise in real life.