فهرست مطالب

Mathematical Modeling - Volume:8 Issue: 4, Summer 2020
  • Volume:8 Issue: 4, Summer 2020
  • تاریخ انتشار: 1399/06/16
  • تعداد عناوین: 7
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  • Marzieh Dehghani-Madiseh * Pages 345-361

    Generalized intervals (intervals whose bounds are not constrained to be increasingly ordered) extend classical intervals and present algebraic completion of conventional interval arithmetic, allowing efficient solution for interval linear systems. In this paper, we use the Cholesky decomposition of a symmetric generalized interval matrix ${bf{A}}$ introduced by Zhao et al. (A generalized Cholesky decomposition for interval matrix, Adv. Mat. Res. 479 (2012) 825--828), to construct the algebraic solution of the triangular interval linear system of equations. Also we utilize this decomposition to find inner and outer estimations of the generalized solution set of the symmetric interval linear systems. Finally some numerical experiments and an application in economic are given to show the efficiency of the presented technique.

    Keywords: Interval arithmetic, Kaucher arithmetic, Cholesky decomposition
  • Mohammad Kazemi * Pages 363-376
    In this paper, we propose a two-stage approach for feature selection in varying coefficient models with ultra-high-dimensional predictors. Specifically, we first employ partial correlation coefficient for screening, and then penalized rank regression is applied for dimension-reduced varying coefficient models to further select important predictors and estimate the coefficient functions. Simulation studies are carried out to examine the performance of proposed approach. We also illustrate it by a real data example.
    Keywords: Big data, feature screening, partial correlation, rank regression
  • Sandeep P. Bhairat * Pages 377-391
    In this paper, we consider a singular differential equation involving Hilfer-Katugampola fractional derivative with the weighted initial condition. The Picard iterative technique has been successfully applied to obtain the existence of a unique solution. First, we derive an equivalent integral equation, then construct the successive approximations and use the ratio test to discuss its convergence. We demonstrate our results through a suitable illustrative example.
    Keywords: Fractional integrals, derivatives, Picard iterative technique, singular fractional differential equation, Cauchy-type problem
  • Hanan A. Wahash *, Satish K. Panchal, Mohammed S. Abdo Pages 393-414
    This article is devoted to the study of a new class of nonlinear fractional-order differential equations with integral boundary conditions involving a generalized version of the Caputo type fractional derivative with respect to another function $h$. In such a path, we transform the proposed problem into an equivalent integral equation. Then we build the upper and lower control functions of the nonlinear term without any monotone requirement except the continuity. By utilizing the method of upper and lower solutions, the fixed point theorems of Schauder and Banach, we obtain the existence and uniqueness of positive solutions for the problem at hand. Finally, we present some examples to illuminate our results.
    Keywords: Caputo fractional differential equation, integral boundary condition, existence of positive solution, control functions, Fixed point theorem
  • Mohammad Nabati *, Soudabeh Nikmanesh Pages 415-433
    In this study, improved Sinc-Galerkin and Sinc-collocation methods are developed based on double exponential transformation to solve a one-dimensional Bratu-type equation. The properties of these methods are used to reduce the solution of the nonlinear problem to the solution of nonlinear algebraic equations. For simplicity in solving the nonlinear system, a matrix vector form of the nonlinear system is found. The upper bound of the error for the Sinc-Galerkin is determined. Also the numerical approximations are compared with the best results reported in the literature. The results confirm that both the Sinc-Galerkin and the Sinc-collocation methods have the same accuracy, but they are significantly more accurate than the other existing methods.
    Keywords: Sinc-Galerkin, Sinc-collocation, Bratu's problem, double exponential transformation, boundary value problems
  • Parisa Rahimkhani, Yadollah Ordokhani * Pages 435-454

    This paper presents an approximate method to solve a class of fractional partial differential equations (FPDEs). First, we introduce   radial basis functions (RBFs) combined with wavelets.  Next, we obtain fractional integral operator (FIO) of wavelets-radial basis functions (W-RBFs) directly.  In the next step, the W-RBFs and their FIO  are used to transform the problem under consideration into a  system of algebraic equations, which can be simply solved to achieve the solution of the problem.   Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the method.

    Keywords: Fractional partial differential equations, radial basis functions, Legendre wavelets, numerical method, fractional integral operator
  • Saadeddine Essarrout *, Said Raghay, Zouhir Mahani Pages 455-477
    In this paper, we study the pharmacokinetics equation for cisplatin (PKC) with random coefficients and initial conditions using the Stochastic Collocation method. We analyze the regularity of the solution with respect to the random variables.   The error estimate for the Stochastic Collocation method is proved using the regularity result and the error estimate for the Finite Difference method. Then, we provide the overall errors estimate and convergence is achieved as a direct result. Some numerical results are simulated to illustrate the theoretical analysis. We also propose a comparison between the stochastic and determinate solving process of PKC equation where we show the efficiency of our adopted method.
    Keywords: Pharmacokinetics (PK) equation for cisplatin, stochastic collocation, Finite difference method, uncertainty quantification