PSI - Issue 28

L.V. Stepanova et al. / Procedia Structural Integrity 28 (2020) 2277–2282 Author name / Structural Integrity Procedia 00 (2019) 000–000

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this approach has been little used in steady state creep – possibly owing to the attractiveness of the energy methods – even though it is the natural approach when using, for instance, finite element method. Nowadays this technique takes on a special significance because of the need and significance of approximate analytical solutions. Modern computer technologies and algorithms provide fast and accurate numerical solutions of complex systems of nonlinear ordinary differential equations. Nevertheless, the problem of constructing analytical approximate solutions of nonlinear problems becomes even more urgent at the present time (Polyanin, (2019), Stepanova and Yakovleva (2015), Singh et al. (2020), Magagula, et al. (2020), Stepanova and Yakovleva (2016), Verma and Tiwari (2019), Rani and Mishra (2020)). Apparently, a reasonable combination of computer technologies and approximate solutions or reductions of nonlinear systems of differential equations gives a promising way of studying physical, economic, biological problems and other problems of natural science. Nowadays the quasilinearization method is promising approach, in the framework of which the solution is constructed through skillful combinations of methods of linear approximation and the use of possibilities modern computing systems. The quasilinearization approach is based on the Newton-Raphson method proposed by Bellman and Kalaba (1965) to solve the nonlinear ordinary and partial differential equations. The quasilinearization method is essentially a generalized Newton-Raphson method for functional equations (Aznam et al. (2019)). It inherits two important properties of the method, namely quadratic convergence and often monotone convergence. The quasilinearization technique linearized the nonlinear boundary value problem and provides a sequence of functions which in general converges quadratically to the solution of the original equation, if there is convergence at all and often has monotone convergence (Bellman and Kalaba (1965), Boyle and Spence (1983), Lakshmikantham and Vatsala (1998)). The solution of the original nonlinear boundary value problem can be obtained through a sequence of successive iterations of the dependent variable. In (Polyanin (2019)) a new direct method for constructing functional separable solutions to non-linear equations of mathematical physics is described. The solutions are sought in the form of an implicit relation containing several free functions (these functions are determined in the subsequent analysis). Different classes of non-linear reaction–diffusion equations with variable coefficients, which admit exact solutions, have been considered. In (Polyanin (2019)) special attention has been paid to non-linear reaction– diffusion equations of general form, which depend on one or several arbitrary functions. Many new generalized traveling-wave solutions and functional separable solutions have been obtained. In (Lakshmikantham and Vatsala (1998)) ideas, basics and systematic developments of the theory of the generalized quasilinearization are presented. The usefulness of the method of quasilinearisation is proved to be very effective for nonlinear problems. Boyle and Spence (1983) provided basic methods of stress analysis for creep and utilized the quasilinearization method for a wide class of nonlinear creep problems. In (Singh et al. (2020)) an efficient method for solving the nonlinear Emden-Fowler type boundary value problems with Dirichlet and Robin-Neumann boundary conditions is introduced. The present method is based on the Haar-wavelets and quasilinearization technique. The quasilinearization technique is adopted to linearize the nonlinear singular problem. Numerical solution of linear singular problem is obtained by the Haar wavelet method (Singh et al. (2020)). The numerical study is further supported by examining thoroughly the convergence of the Haar wavelet method and the quasilinearization technique. In order to check the accuracy of the proposed method, the numerical results are compared with both existing methods and exact solutions. In (Magagula et al. (2020)) a new innovative technique for solving nonlinear systems of partial differential equations with quadratic convergence rate is presented. The use of the method by solving a nonlinear system of partial differential equations that belong to a class of nonsimilar boundary layer equations is demonstrated. The model equations describe the problem of magnetohydrodynamic forced convection flow adjacent to a nonisothermal wedge. Convergence analysis and grid independence tests are conducted to establish the accuracy, convergence, and validity of the proposed method. Computational order of convergence is proved numerically and is presented in a tabular form. The results are also validated against published results. The method is found to perform better than existing methods for a class of nonsimilar boundary layer equations, converges faster, and uses fewer grid points to achieve accurate results. The method uses minimal computation time, and we show that the accuracy of the method does not deteriorate for large values of the governing independent variables. Nomenclature ij  stress tensor components , r  in-plane coordinates ij   creep strain rates , i ij a b coefficients in the linearization method n creep exponent