Solving nonlinear boundary value problems by a boundary shape function method and a splitting and linearizing method

In the paper, we develop two novel iterative methods to determine the solution of a second-order nonlinear boundary value problem (BVP), which precisely satisfies the specified non-separable boundary conditions by taking advantage of the property of the corresponding boundary shape function (BSF). The first method based on the BSF can exactly transform the BVP to an initial value problem for the new variable with two given initial values, while two unknown terminal values are determined iteratively. By using the BSF in the second method, we derive the fractional powers exponential functions as the bases, which automatically satisfy the boundary conditions. A new splitting and linearizing technique is used to transform the nonlinear BVP into linear equations at each iteration step, which are solved to determine the expansion coefficients and then the solution is available. Upon adopting those two novel methods very accurate solution for the nonlinear BVP with non-separable boundary conditions can be found quickly. Several numerical examples are solved to assess the efficiency and accuracy of the proposed iterative algorithms, which are compared to the shooting method.

Qualitative analysis of magnetohydrodynamics stagnation point flow with a novel numerical approach through Shifted Chebyshev collocation method

This paper investigates the third-order nonlinear boundary value problem, resulting from the exact reduction of the Navier-Stokes equation caused by the magnetohydrodynamics boundary layer flow near a stagnation point on a rough plate. The governing partial differential equations are transformed into a nonlinear ordinary differential equation and partial slip boundary conditions by an appropriate similarity transformation. In this previous work, the boundary value problem (BVP) was investigated numerically, and a lot of speculations regarding the existence and behavior of the solutions were carried out. The primary objective of this article is to verify these speculations mathematically. In this work, we have proved that there is a unique solution for all parameters values, and further, the solution has monotonic increasing first derivative. Moreover, the resulting nonlinear boundary value problem is solved by shifted Chebyshev collocation method. We compare the present numerical results with the previous results for the particular physical parameters, concluding that the results are highly accurate. The velocity profiles and streamlines are also plotted to address the significance of the parameters. Our manuscript is a judicial mix between mathematical and numerical methods.

Methods for solving the nonlinear equilibrium equation of a compressed elastic rod

A nonlinear boundary value problem on the equilibrium of a compressed elastic rod on nonlinear foundation is considered for cases of free pinching or pivotally supported of the ends. The problem is written as a nonlinear operator equation. Numerical and analytical methods for solving nonlinear boundary value problems are discussed: The Newton-Kantorovich method and the Lyapunov-Schmidt method. We also consider a problem linearized on a trivial solution (the eigenvalue problem), which has an exact solution (Euler) in the case of a hinge support, and for the case of pinching the ends of the rod, the solution formulas are obtained in the works of A. A. Esipov and V. I. Yudovich. The eigenvalue problem is also solved by numerical method. To determine the equilibria of a nonlinear boundary value problem for a given value of the compressive force, it is proposed to apply the Newton-Kantorovich method in combination with the numerical methods, using as initial approximations the asymptotic formulas of new solutions found using the Lyapunov-Schmidt method in the vicinity of the critical value closest to the current value of the compressive load. Numerical calculations are performed and conclusions are drawn about the effectiveness of the methods used.

Modeling of a temperature field for extruder body

The paper considers the process of induction heating of the extruder body, the temperature of which determines the degree of heating of the polymer mixture in the zone of loading the dry mixture. A mathematical model of this process is formulated taking into account radiant heat transfer in the gap between the inductor and the case. An iterative numerical-analytical method is proposed for solving the corresponding nonlinear boundary value problem of housing heating, at the first iteration of which a linear boundary value problem is solved (without taking into account radiant heat transfer). At the subsequent stages, a nonlinear boundary value problem is solved. The iterative method is based on the application of integral transformations of the linear part of the problem, followed by an iterative scheme for finding a nonlinear problem. This scheme is based on the algorithms for the equivalent simplification of the expressions obtained by solving the problem. The results of mathematical modeling of the corresponding algorithms are presented.