Approximating Solutions of Non-Linear Troesch’s Problem via an Efficient Quasi-Linearization Bessel Approach

Two collocation-based methods utilizing the novel Bessel polynomials (with positive coefficients) are developed for solving the non-linear Troesch’s problem. In the first approach, by expressing the unknown solution and its second derivative in terms of the Bessel matrix form along with some collocation points, the governing equation transforms into a non-linear algebraic matrix equation. In the second approach, the technique of quasi-linearization is first employed to linearize the model problem and, then, the first collocation method is applied to the sequence of linearized equations iteratively. In the latter approach, we require to solve a linear algebraic matrix equation in each iteration. Moreover, the error analysis of the Bessel series solution is established. In the end, numerical simulations and computational results are provided to illustrate the utility and applicability of the presented collocation approaches. Numerical comparisons with some existing available methods are performed to validate our results.

A new multistage Parker-Sochacki method for solving the Troesch’s problem

In this article, we introduce a new method to obtain an approximate analytical solution of the highly unstable Troesch’s problem. In the proposed method, without recourse to any hyperbolic tangent transformation or finite term approximation of the hyperbolic sine function, the problem is recast as a system of projectively polynomials which allows straightforward computation of the series solution of the problem. The radius of convergence of the series solution to the problem is derived a-priorly in terms of the parameters of the polynomial system. Using a step length ; the problem domain is divided into subintervals, where corresponding subproblems are defined and solved with Parker-Sochacki method with very high accuracy. Highly accurate piecewise continuous approximate solution is thus obtained on the entire integration interval. The obtained solution, which is valid for every choice of the Troesch parameter , showed comparable accuracy to known numerical solutions in the literature. In particular, new results are presented for large values of in the range [20;500].

An easy and computable approximation for Troesch’s problem by using the Laplace Transform-Homotopy Perturbation Method

This work introduces the Laplace Transform-Homotopy Perturbation Method (LT-HPM) in order to provide an approximate solution for Troesch’s problem. After comparing figures between exact and approximate solutions, as well as the average absolute relative error (AARE) of the approximate solutions of this research, with others reported in the literature, it can be said that the proposed solutions are accurate and handy. In conclusion, LT-HPM is a potentially useful tool.

SOLVING TROESCH’S PROBLEM BY USING MODIFIED NONLINEAR SHOOTING METHOD

In this research article, the non-linear shooting method is modified (MNLSM) and is considered to simulate Troesch’s sensitive problem (TSP) numerically. TSP is a 2nd order non-linear BVP with Dirichlet boundary conditions. In MNLSM, classical 4th order Runge-Kutta method is replaced by Adams-Bashforth-Moulton method, both for systems of ODEs. MNLSM showed to be efficient and is easy for implementation. Numerical results are given to show the performance of MNLSM, compared to the exact solution and to the results by He’s polynomials. Also, discussion of results and the comparison with other applied techniques from the literature are given for TSP.