Search results for: quasilinearization

Authors: Pradeepa Teegala, Ramreddy Chetteti

Abstract:

This work emphasizes the effects of heat generation/absorption and Joule heating on chemically reacting micropolar fluid flow over a truncated cone with convective boundary condition. For this complex fluid flow problem, the similarity solution does not exist and hence using non-similarity transformations, the governing fluid flow equations along with related boundary conditions are transformed into a set of non-dimensional partial differential equations. Several authors have applied the spectral quasi-linearization method to solve the ordinary differential equations, but here the resulting nonlinear partial differential equations are solved for non-similarity solution by using a recently developed method called the spectral quasi-linearization method (SQLM). Comparison with previously published work on special cases of the problem is performed and found to be in excellent agreement. The influence of pertinent parameters namely Biot number, Joule heating, heat generation/absorption, chemical reaction, micropolar and magnetic field on physical quantities of the flow are displayed through graphs and the salient features are explored in detail. Further, the results are analyzed by comparing with two special cases, namely, vertical plate and full cone wherever possible.

Keywords: chemical reaction, convective boundary condition, joule heating, micropolar fluid, spectral quasilinearization method

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Authors: K. N. S. Kasi Viswanadham

Abstract:

Coupled system of second order linear and nonlinear boundary value problems occur in various fields of Science and Engineering. In the formulation of the problem, any one of 81 possible types of boundary conditions may occur. These 81 possible boundary conditions are written as a combination of four boundary conditions. To solve a coupled system of boundary value problem with these converted boundary conditions, a collocation method with cubic B-splines as basis functions has been developed. In the collocation method, the mesh points of the space variable domain have been selected as the collocation points. The basis functions have been redefined into a new set of basis functions which in number match with the number of mesh points in the space variable domain. The solution of a non-linear boundary value problem has been obtained as the limit of a sequence of solutions of linear boundary value problems generated by quasilinearization technique. Several linear and nonlinear boundary value problems are presented to test the efficiency of the proposed method and found that numerical results obtained by the present method are in good agreement with the exact solutions available in the literature.

Keywords: collocation method, coupled system, cubic b-splines, mesh points

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Authors: Kizito Ugochukwu Nwajeria

Abstract:

This paper presents a spectral quasi-linearization technique (SQLT) for solving time fractional diffusion wave equations in boundary value problems. The proposed method integrates spectral approximations for spatial derivatives with a quasi-linearization approach to address the nonlinearity introduced by fractional time derivatives. Time fractional differential equations typically formulated using Caputo or Riemann-Liouville derivatives, model complex phenomena such as anomalous diffusion and wave propagation, which are not captured by classical integer-order models. The SQLT method iteratively linearizes the nonlinear terms at each time step, transforming the original problem into a series of linear subproblems, which can be efficiently solved. Using high-order spectral methods such as Chebyshev or Legendre polynomials for spatial discretization, the technique achieves high accuracy in approximating the solution. A convergence analysis is provided, demonstrating the method's efficiency and establishing error bounds. Numerical experiments on a range of test problems confirm the effectiveness of SQLT in solving fractional diffusion wave equations with various boundary conditions. The method offers a robust framework for addressing time fractional differential equations in diverse fields, including materials science, bioengineering, and anomalous transport phenomena.

Keywords: spectral methods, quasilinearization, time-fractional diffusion-wave equations, boundary value problems, fractional calculus

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