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 M NLSM , classical 4 th 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

A Modification and Application of Parametric Continuation Method to Variety of Nonlinear Boundary Value Problems in Applied Mechanics

In the field of engineering, researches often come across strong nonlinear boundary value problems which cannot be solved easily. Numerical convergence for many problems, typically solved by the Newton-Raphson linearization algorithm, is sensitive to the initial approach, relaxation parameters and differential topology. Emphasis in the present work is placed on the alternative approach, the so called parametric imbedding of a particular problem into the family of problems. While this may appear to complicate rather than to simplify the problem, its justification lies in the fact that a relation between infinitesimally close neighboring processes results in a simple Cauchy problem with respect to the introduced parameter. Many problems in applied mechanics are reduced to the solutions of systems of nonlinear algebraic, transcendental, differential or integral-differential equations containing an explicit parameter. These are problems in the areas of thermo-fluids, gas dynamics, deformable solids, heat transfer, biomechanics, analytical dynamics, catastrophe theory, optimal control and others. A parameter found in these models is not unique, and may be easily identified as a load which could be geometric, structural, and physical or it could be introduced artificially. An important aspect of these problems is a question of the variation of the solution when parameter is incrementally changed. The growing interest in nonlinear problems in engineering has been intensified by the use of digital computers. This paved a way in development of the solution procedures which can be applied to a large class of nonlinear problems containing a parameter. An important aspect of these problems is the variation of the solution of with the parameter. Hence, method of continuing the solution with respect to the parameter is a natural and universal tool for the analysis. It was originally introduced by Ambarzumian and Chandrasekar, and intensively studied by Bellman, Kalaba and others. Different problems of applied mechanics and physics with dominant nonlinearities due to convective phenomena, constituent models, finite deformation, bifurcation and others are analyzed and solved in the present work. The choice of the optimal continuation parameter, which ensures the best conditioning of the corresponding system of nonlinear equations, is discussed. Some modifications for stiff systems of ordinary nonlinear differential equations are suggested and applied. Effectiveness of the continuation method is demonstrated by comparing the results with the stiff boundary value problem numerical solvers implemented using commercial softwares. The objective of the research is to investigate applicability of the method as a universal approach to the wide range of nonlinear boundary value problems in different areas of mechanics: nonlinear mechanics of solids, bifurcation problems, Newtonian and Non-Newtonian fluids, thermo-fluids, gas-dynamics, control, inverse problems

An Improved Training Method For Wavelet Neural Networks For Solving Ordinary And Partial Differential Equations

Differential equations (DEs) have been widely used for modelling countless studies and applications in various disciplines. In recent years, artificial neural networks (ANNs) have gained attention in solving DEs in a more efficient way to approximate solutions of DEs. In this thesis, wavelet neural networks (WNNs) were employed and programmed to solve ordinary differential equations (ODEs) and partial differential equations (PDEs)