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  • Open Access


    Computational Methods for Non-Linear Equations with Some Real-World Applications and Their Graphical Analysis

    Amir Naseem1, M.A. Rehman1, Thabet Abdeljawad2,3,4,*

    Intelligent Automation & Soft Computing, Vol.30, No.3, pp. 805-819, 2021, DOI:10.32604/iasc.2021.019164

    Abstract In this article, we propose some novel computational methods in the form of iteration schemes for computing the roots of non-linear scalar equations in a new way. The construction of these iteration schemes is purely based on exponential series expansion. The convergence criterion of the suggested schemes is also given and certified that the newly developed iteration schemes possess quartic convergence order. To analyze the suggested schemes numerically, several test examples have been given and then solved. These examples also include some real-world problems such as van der Wall’s equation, Plank’s radiation law and kinetic problem equation whose numerical results… More >

  • Open Access


    Solution of Post-Buckling & Limit Load Problems, Without Inverting the Tangent Stiffness Matrix & Without Using Arc-Length Methods

    T.A. Elgohary1, L. Dong2, J.L. Junkins3, S.N. Atluri4

    CMES-Computer Modeling in Engineering & Sciences, Vol.98, No.6, pp. 543-563, 2014, DOI:10.3970/cmes.2014.098.543

    Abstract In this study, the Scalar Homotopy Methods are applied to the solution of post-buckling and limit load problems of solids and structures, as exemplified by simple plane elastic frames, considering only geometrical nonlinearities. Explicitly derived tangent stiffness matrices and nodal forces of large-deformation planar beam elements, with two translational and one rotational degrees of freedom at each node, are adopted following the work of [Kondoh and Atluri (1986)]. By using the Scalar Homotopy Methods, the displacements of the equilibrium state are iteratively solved for, without inverting the Jacobian (tangent stiffness) matrix. It is well-known that, the simple Newton’s method (and… More >

  • Open Access


    A Scalar Homotopy Method with Optimal Hybrid Search Directions for Solving Nonlinear Algebraic Equations

    Weichung Yeih1,2, Cheng-Yu Ku1,2,3, Chein-Shan Liu4, I-Yao Chan1,2

    CMES-Computer Modeling in Engineering & Sciences, Vol.90, No.4, pp. 255-282, 2013, DOI:10.3970/cmes.2013.090.255

    Abstract In this paper, a scalar homotopy method with optimal hybrid search directions for solving nonlinear algebraic equations is proposed. To conduct the proposed method, we first convert the vector residual function to a scalar function by taking the square norm of the vector function and then, introduce a fictitious time variable to form a scalar homotopy function. To improve the convergence and the accuracy of the proposed method, a vector with multiple search directions and an iterative algorithm are introduced into the evolution dynamics of the solutions. Further, for obtaining the optimal search direction, linear and nonlinear optimization algorithms are… More >

  • Open Access


    Simple "Residual-Norm" Based Algorithms, for the Solution of a Large System of Non-Linear Algebraic Equations, which Converge Faster than the Newton’s Method

    Chein-Shan Liu1, Satya N. Atluri2

    CMES-Computer Modeling in Engineering & Sciences, Vol.71, No.3, pp. 279-304, 2011, DOI:10.3970/cmes.2011.071.279

    Abstract For solving a system of nonlinear algebraic equations (NAEs) of the type: F(x)=0, or Fi(xj) = 0, i,j = 1,...,n, a Newton-like algorithm has several drawbacks such as local convergence, being sensitive to the initial guess of solution, and the time-penalty involved in finding the inversion of the Jacobian matrix ∂Fi/∂xj. Based-on an invariant manifold defined in the space of (x,t) in terms of the residual-norm of the vector F(x), we can derive a gradient-flow system of nonlinear ordinary differential equations (ODEs) governing the evolution of x with a fictitious time-like variable t as an independent variable. We can prove… More >

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