Kazunori Shinohara^*

CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.2, pp. 441-473, 2020, DOI:10.32604/cmes.2020.08656 - 01 May 2020

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, … More >

T. Binesh^{1, *}

CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.1, pp. 333-351, 2020, DOI:10.32604/cmes.2020.08097 - 01 April 2020

Abstract Wavelength-dependent mathematical modelling of the differential energy change of a photon has been performed inside a proposed hypothetical optical medium. The existence of this medium demands certain mathematical constraints, which have been derived in detail. Using reverse modelling, a medium satisfying the derived conditions is proven to store energy as the photon propagates from the entry to exit point. A single photon with a given intensity is considered in the analysis and hypothesized to possess a definite non-zero probability of maintaining its energy and velocity functions analytic inside the proposed optical medium, despite scattering, absorption, More >

W. M. Abd-Elhameed^1,2,*, Y. H. Youssri²

CMES-Computer Modeling in Engineering & Sciences, Vol.121, No.3, pp. 1029-1049, 2019, DOI:10.32604/cmes.2019.08378

Abstract This paper is confined to analyzing and implementing new spectral solutions of the fractional Riccati differential equation based on the application of the spectral tau method. A new explicit formula for approximating the fractional derivatives of shifted Chebyshev polynomials of the second kind in terms of their original polynomials is established. This formula is expressed in terms of a certain terminating hypergeometric function of the type ₄F₃(1). This hypergeometric function is reduced in case of the integer case into a certain terminating hypergeometric function of the type ₃F₂(1) which can be summed with the aid of More >

Mahdi Saedshoar Heris¹, Mohammad Javidi¹, Bashir Ahmad^2,*

CMES-Computer Modeling in Engineering & Sciences, Vol.121, No.1, pp. 249-272, 2019, DOI:10.32604/cmes.2019.08080

Abstract In this paper, we propose numerical methods for the Riesz space fractional advection-dispersion equations with delay (RFADED). We utilize the fractional backward differential formulas method of second order (FBDF2) and weighted shifted Grünwald difference (WSGD) operators to approximate the Riesz fractional derivative and present the finite difference method for the RFADED. Firstly, the FBDF2 and the shifted Grünwald methods are introduced. Secondly, based on the FBDF2 method and the WSGD operators, the finite difference method is applied to the problem. We also show that our numerical schemes are conditionally stable and convergent with the accuracy More >

Kazunori Shinohara^{1, *}

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… More >

Yanxin Wang^{1, *}, Li Zhu¹, Zhi Wang¹

CMES-Computer Modeling in Engineering & Sciences, Vol.118, No.2, pp. 339-350, 2019, DOI:10.31614/cmes.2019.04575

Abstract An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper. The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented. Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations. And the convergence of the Euler wavelets basis is given. The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy. More >

Muhammad Shoaib Arif^1,*, Ali Raza¹, Muhammad Rafiq², Mairaj Bibi³, Rabia Fayyaz³, Mehvish Naz³, Umer Javed⁴

CMC-Computers, Materials & Continua, Vol.59, No.3, pp. 787-804, 2019, DOI:10.32604/cmc.2019.04655

Abstract In this paper, a reliable stochastic numerical analysis for typhoid fever incorporating with protection against infection has been considered. We have compared the solutions of stochastic and deterministic typhoid fever model. It has been shown that the stochastic typhoid fever model is more realistic as compared to the deterministic typhoid fever model. The effect of threshold number T* hold in stochastic typhoid fever model. The proposed framework of the stochastic non-standard finite difference scheme (SNSFD) preserves all dynamical properties like positivity, bounded-ness and dynamical consistency defined by Mickens, R. E. The stochastic numerical simulation of More >

Timon Rabczuk^1,2,*, Huilong Ren³, Xiaoying Zhuang^4,5

CMC-Computers, Materials & Continua, Vol.59, No.1, pp. 31-55, 2019, DOI:10.32604/cmc.2019.04567

Abstract A novel nonlocal operator theory based on the variational principle is proposed for the solution of partial differential equations. Common differential operators as well as the variational forms are defined within the context of nonlocal operators. The present nonlocal formulation allows the assembling of the tangent stiffness matrix with ease and simplicity, which is necessary for the eigenvalue analysis such as the waveguide problem. The present formulation is applied to solve the differential electromagnetic vector wave equations based on electric fields. The governing equations are converted into nonlocal integral form. An hourglass energy functional is More >

Ali Raza¹, Muhammad Shoaib Arif^1,*, Muhammad Rafiq², Mairaj Bibi³, Muhammad Naveed¹, Muhammad Usman Iqbal⁴, Zubair Butt⁴, Hafiza Anum Naseem⁴, Javeria Nawaz Abbasi³

CMES-Computer Modeling in Engineering & Sciences, Vol.120, No.2, pp. 445-465, 2019, DOI:10.32604/cmes.2019.06454

Abstract This writing is an attempt to explain a reliable numerical treatment for stochastic computer virus model. We are comparing the solutions of stochastic and deterministic computer virus models. This paper reveals that a stochastic computer virus paradigm is pragmatic in contrast to the deterministic computer virus model. Outcomes of threshold number C^∗ hold in stochastic computer virus model. If C^∗ < 1 then in such a condition virus controlled in the computer population while C^∗ > 1 shows virus persists in the computer population. Unfortunately, stochastic numerical methods fail to cope with large step sizes of time. More >

Shixiong Wang^1,∗, Jianhua He¹, Chen Wang², Xitong Li¹

Computer Systems Science and Engineering, Vol.33, No.5, pp. 379-387, 2018, DOI:10.32604/csse.2018.33.379

Abstract Many Engineering Problems could be mathematically described by FinalValue Problem, which is the inverse problem of InitialValue Problem. Accordingly, the paper studies the final value problem in the field of ODE problems and analyses the differences and relations between initial and final value problems. The more general new concept of the endpoints-value problem which could describe both initial and final problems is proposed. Further, we extend the concept into inner-interval value problem and arbitrary value problem and point out that both endpoints-value problem and inner-interval value problem are special forms of arbitrary value problem. Particularly, More >