The general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated. The generalization of the proof to partial differential equations is straight forward. The method, whose mathematical basis in physics was discussed recently by one of the present ...

Orthogonal functions and polynomials have been used by many authors for solving various problems. The main idea of using orthogonal basis is that a problem reduces to solving a system of linear or nonlinear algebraic equations by truncated series of orthogonal basis functions for solution of problem and using the operational matrices. Here we use Legendre wavelets basis on interval [0, 1]. Some...

The wavelets have many applications in engineering and the sciences, especially mathematics. Recently, 2021, wavelet Boubaker (WB) polynomials were used for first time to study their properties detail. They also utilized solving Lane-Emden equation. aim of this paper is show truncated Wavelet variation problems. In research, direct method using was presented variational reduces problem into a s...

Numerical solutions of the isothermal Lane-Emden equation are presented, corresponding to self-gravitating gaseous cores embedded within a finite density envelope of overall cylindrical symmetry. These structures may be members of a fragmentation hierarchy proceeding from sheets, to filaments, to elongated, prolate clumps. The embedded solutions are the first of their kind, and as such represen...

We are concerned with the Lane-Emden problem{−Δu=upinΩ,u>0inΩ,u=0on∂Ω, where Ω⊂R2 is a smooth bounded domain and p>1 sufficiently large. Improving some known asymptotic estimates on solutions, we prove non-degeneracy local uniqueness of multi-spikes positive solutions for general domains. Our methods mainly use ODE's theory, various Pohozaev identities, blow-up analysis properties Green's funct...