Establishing existence and uniqueness of solutions to the boundary value problem involving a generalized Emden equation,
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Establishing existence and uniqueness of solutions to the boundary value problem involving a generalized Emden equation, embracing Thomas–Fermi-like theories Saleh S. Almuthaybiri
· Christopher C. Tisdell
Received: 27 June 2019 / Accepted: 29 May 2020 / Published online: 24 September 2020 © Springer Nature B.V. 2020
Abstract The purpose of this article is to construct a firm mathematical foundation for the boundary value problem associated with a generalized Emden equation that embraces Thomas–Fermi-like theories. Boundary value problems for the relativistic and non-relativistic Thomas–Fermi equations are included as special cases. Questions of existence and uniqueness of solutions to these boundary value problems form a fundamental and important area of investigation regarding whether these mathematical models for physical phenomena are actually well-posed. However, these questions have remained open for the generalized Emden problem and the relativistic Thomas–Fermi problem. Herein we advance current understanding of existence and uniqueness of solutions by proving that these boundary value problems each admit a unique solution. Our methods involve an analysis of the problems through arguments that apply differential inequalities and fixed-point theory. The new results guarantee the existence of a unique solution, ensuring the generalized Emden equation that embraces Thomas–Fermi theory sits on a firm mathematical foundation. Keywords Boundary value problem · Emden equation · Existence · Thomas–Fermi equation · Uniqueness Mathematics Subject Classification 34B60 · 34B15
1 Introduction The evolution of the Thomas–Fermi theory [1,2] of atoms has captivated the scientific attention of research communities in applied mathematics and physics for in excess of eighty years. There is a vast amount of literature regarding the field and a detailed review of it is beyond the scope of the present work, but for example, see [3–29] and the references therein. S. S. Almuthaybiri · Christopher C. Tisdell (B) School of Mathematics & Statistics, The University of New South Wales, UNSW, Sydney 2052, Australia e-mail: [email protected] S. S. Almuthaybiri Department of Mathematics, College of Science and Arts in Uglat Asugour, Qassim University, Buraydah, Kingdom of Saudi Arabia e-mail: [email protected]
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S. S. Almuthaybiri, C. C. Tisdell
The purpose of the present work is to establish a firm mathematical foundation for the non-linear differential equation n x 2n−1 y = x y + λy 2
(1.1)
where n and λ are constants and (1.1) is subjected to the (Dirichlet) boundary conditions y(0) = 1,
y(x0 ) = 0, x0 > 0.
(1.2)
The general form (1.1) can be linked with multiple models that are of physical interest and we briefly discuss some special cases to help motivate and contextualize our study. The case n = 3/2 and λ = 0 in (1.1) leads to the differential equation y =
y 3/2 , x 1/2
(1.3)
which is known as the classic, dimensionless Thomas–Fermi equation [1,2]. For an almost encyclopedic account of the literature
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