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  • Open Access

    ARTICLE

    An Effective Numerical Method for the Solution of a Stochastic Coronavirus (2019-nCovid) Pandemic Model

    Wasfi Shatanawi1,2,3, Ali Raza4,5,*, Muhammad Shoaib Arif4, Kamaledin Abodayeh1, Muhammad Rafiq6, Mairaj Bibi7

    CMC-Computers, Materials & Continua, Vol.66, No.2, pp. 1121-1137, 2021, DOI:10.32604/cmc.2020.012070

    Abstract Nonlinear stochastic modeling plays a significant role in disciplines such as psychology, finance, physical sciences, engineering, econometrics, and biological sciences. Dynamical consistency, positivity, and boundedness are fundamental properties of stochastic modeling. A stochastic coronavirus model is studied with techniques of transition probabilities and parametric perturbation. Well-known explicit methods such as Euler Maruyama, stochastic Euler, and stochastic Runge–Kutta are investigated for the stochastic model. Regrettably, the above essential properties are not restored by existing methods. Hence, there is a need to construct essential properties preserving the computational method. The non-standard approach of finite difference is examined to maintain the above basic… More >

  • Open Access

    ARTICLE

    Residual Correction Procedure with Bernstein Polynomials for Solving Important Systems of Ordinary Differential Equations

    M. H. T. Alshbool1, W. Shatanawi2, 3, 4, *, I. Hashim5, M. Sarr1

    CMC-Computers, Materials & Continua, Vol.64, No.1, pp. 63-80, 2020, DOI:10.32604/cmc.2020.09431

    Abstract One of the most attractive subjects in applied sciences is to obtain exact or approximate solutions for different types of linear and nonlinear systems. Systems of ordinary differential equations like systems of second-order boundary value problems (BVPs), Brusselator system and stiff system are significant in science and engineering. One of the most challenge problems in applied science is to construct methods to approximate solutions of such systems of differential equations which pose great challenges for numerical simulations. Bernstein polynomials method with residual correction procedure is used to treat those challenges. The aim of this paper is to present a technique… More >

  • Open Access

    ARTICLE

    Addition Formulas of Leaf Functions and Hyperbolic Leaf Functions

    Kazunori Shinohara*

    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.2, pp. 441-473, 2020, DOI:10.32604/cmes.2020.08656

    Abstract Addition formulas exist in trigonometric functions. Double-angle and half-angle formulas can be derived from these formulas. Moreover, the relation equation between the trigonometric function and the hyperbolic function can be derived using an imaginary number. The inverse hyperbolic function is similar to the inverse trigonometric function , such as the second degree of a polynomial and the constant term 1, except for the sign − and +. Such an analogy holds not only when the degree of the polynomial is 2, but also for higher degrees. As such, a function exists with respect… More >

  • Open Access

    ARTICLE

    Damped and Divergence Exact Solutions for the Duffing Equation Using Leaf Functions and Hyperbolic Leaf Functions

    Kazunori Shinohara1, *

    CMES-Computer Modeling in Engineering & Sciences, Vol.118, No.3, pp. 599-647, 2019, DOI:10.31614/cmes.2019.04472

    Abstract According to the wave power rule, the second derivative of a function x(t) with respect to the variable t is equal to negative n times the function x(t) raised to the power of 2n-1. Solving the ordinary differential equations numerically results in waves appearing in the figures. The ordinary differential equation is very simple; however, waves, including the regular amplitude and period, are drawn in the figure. In this study, the function for obtaining the wave is called the leaf function. Based on the leaf function, the exact solutions for the undamped and unforced Duffing equations are presented. In the… More >

  • Open Access

    ARTICLE

    Nonstandard Group-Preserving Schemes for Very Stiff Ordinary Differential Equations

    Chein-Shan Liu1

    CMES-Computer Modeling in Engineering & Sciences, Vol.9, No.3, pp. 255-272, 2005, DOI:10.3970/cmes.2005.009.255

    Abstract The group-preserving scheme developed by Liu (2001) for calculating the solutions of k-dimensional differential equations system adopted the Cayley transform to formulate the Lie group from its Lie algebra A ∈ so(k,1). In this paper we consider a more effective exponential mapping to derive exp(hA). In order to overcome the difficulty of numerical instabilities encountered by employing group-preserving schemes on stiff differential equations, we further combine the nonstandard finite difference method into the group-preserving schemes to obtain unconditional stable numerical methods. They provide single-step explicit time integrators for stiff differential equations. Several numerical examples are examined, some of which are… More >

  • Open Access

    ARTICLE

    Efficient Shooting Methods for the Second-Order Ordinary Differential Equations

    Chein-Shan Liu1

    CMES-Computer Modeling in Engineering & Sciences, Vol.15, No.2, pp. 69-86, 2006, DOI:10.3970/cmes.2006.015.069

    Abstract In this paper we will study the numerical integrations of second order boundary value problems under the imposed conditions at t=0 and t=T in a general setting. We can construct a compact space shooting method for finding the unknown initial conditions. The key point is based on the construction of a one-step Lie group element G(u0,uT) and the establishment of a mid-point Lie group element G(r). Then, by imposing G(u0,uT) = G(r) we can search the missing initial conditions through an iterative solution of the weighting factor r ∈ (0,1). Numerical examples were examined to convince that the new approach… More >

  • Open Access

    ARTICLE

    An Iterative Algorithm for Solving a System of Nonlinear Algebraic Equations, F(x) = 0, Using the System of ODEs with an Optimum α in x· = λ[αF + (1−α)BTF]; Bij = ∂Fi/∂xj

    Chein-Shan Liu1, Satya N. Atluri2

    CMES-Computer Modeling in Engineering & Sciences, Vol.73, No.4, pp. 395-432, 2011, DOI:10.3970/cmes.2011.073.395

    Abstract In this paper we solve a system of nonlinear algebraic equations (NAEs) of a vector-form: F(x) = 0. Based-on an invariant manifold defined in the space of (x,t) in terms of the residual-norm of the vector F(x), we derive a system of nonlinear ordinary differential equations (ODEs) with a fictitious time-like variable t as an independent variable: x· = λ[αF + (1−α)BTF], where λ and α are scalars and Bij = ∂Fi/∂xj. From this set of nonlinear ODEs, we derive a purely iterative algorithm for finding the solution vector x, without having to invert the Jacobian (tangent stiffness matrix)… More >

  • Open Access

    ARTICLE

    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 >

  • Open Access

    ARTICLE

    An Enhanced Fictitious Time Integration Method for Non-Linear Algebraic Equations With Multiple Solutions: Boundary Layer, Boundary Value and Eigenvalue Problems

    Chein-Shan Liu1, Weichung Yeih2, Satya N. Atluri3

    CMES-Computer Modeling in Engineering & Sciences, Vol.59, No.3, pp. 301-324, 2010, DOI:10.3970/cmes.2010.059.301

    Abstract When problems in engineering and science are discretized, algebraic equations appear naturally. In a recent paper by Liu and Atluri, non-linear algebraic equations (NAEs) were transformed into a nonlinear system of ODEs, which were then integrated by a method labelled as the Fictitious Time Integration Method (FTIM). In this paper, the FTIM is enhanced, by using the concept of arepellorin the theory ofnonlinear dynamical systems, to situations where multiple-solutions exist. We label this enhanced method as MSFTIM. MSFTIM is applied and illustrated in this paper through solving boundary-layer problems, boundary-value problems, and eigenvalue problems with multiple solutions. More >

  • Open Access

    ARTICLE

    A Scalar Homotopy Method for Solving an Over/Under-Determined System of Non-Linear Algebraic Equations

    Chein-Shan Liu1, Weichung Yeih2, Chung-Lun Kuo3, Satya N. Atluri4

    CMES-Computer Modeling in Engineering & Sciences, Vol.53, No.1, pp. 47-72, 2009, DOI:10.3970/cmes.2009.053.047

    Abstract Iterative algorithms for solving a system of nonlinear algebraic equations (NAEs): Fi(xj) = 0, i, j = 1,... ,n date back to the seminal work of Issac Newton. Nowadays a Newton-like algorithm is still the most popular one to solve the NAEs, due to the ease of its numerical implementation. However, this type of algorithm is sensitive to the initial guess of solution, and is expensive in terms of the computations of the Jacobian matrix ∂Fi/∂xj and its inverse at each iterative step. In addition, the Newton-like methods restrict one to construct an iteration procedure for n-variables by using n-equations,… More >

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