Space-time fractional stochastic partial differential equations

Mijena, J.B.,Nane, E. North-Holland Pub. Co ; Elsevier Science Ltd 2015 Stochastic processes and their applications Vol.125 No.9

We consider non-linear time-fractional stochastic heat type equation @?<SUB>t</SUB><SUP>β</SUP>u<SUB>t</SUB>(x)=-ν(-Δ)<SUP>α/2</SUP>u<SUB>t</SUB>(x)+I<SUB>t</SUB><SUP>1-β</SUP>[σ(u)W@?(t,x)] in (d+1) dimensions, where ν>0,β@?(0,1), α@?(0,2] and d<min{2,β<SUP>-1</SUP>}α, @?<SUB>t</SUB><SUP>β</SUP> is the Caputo fractional derivative, -(-Δ)<SUP>α/2</SUP> is the generator of an isotropic stable process, I<SUB>t</SUB><SUP>1-β</SUP> is the fractional integral operator, W@?(t,x) is space-time white noise, and σ:R→R is Lipschitz continuous. Time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. We prove existence and uniqueness of mild solutions to this equation and establish conditions under which the solution is continuous. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in Foondun and Khoshnevisan (2009), Walsh (1986). In sharp contrast to the stochastic partial differential equations studied earlier in Foondun and Khoshnevisan (2009), Khoshnevisan (2014) and Walsh (1986), in some cases our results give existence of random field solutions in spatial dimensions d=1,2,3. Under faster than linear growth of σ, we show that time fractional stochastic partial differential equation has no finite energy solution. This extends the result of Foondun and Parshad (in press) in the case of parabolic stochastic partial differential equations. We also establish a connection of the time fractional stochastic partial differential equations to higher order parabolic stochastic differential equations.

Interval oscillation theorems for second-order differential equations

Zheng Bin 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.3

In this paper,we are concernedwith a class of nonlinear second- order differential equations with a nonlinear damping term and forcing term: (r(t)k1(x(t),x'(t)))' + p(t)k2(x(t),x'(t))x'(t) + q(t)f(x(t)) = 0. Passage to more general class of equations allows us to remove a restrictive condition usually imposed on the nonlinearity. And, as a consequence, our results apply to wider classes of nonlinear differential equations.Some illustrative examples are considered. In this paper,we are concernedwith a class of nonlinear second- order differential equations with a nonlinear damping term and forcing term: (r(t)k1(x(t),x'(t)))' + p(t)k2(x(t),x'(t))x'(t) + q(t)f(x(t)) = 0. Passage to more general class of equations allows us to remove a restrictive condition usually imposed on the nonlinearity. And, as a consequence, our results apply to wider classes of nonlinear differential equations.Some illustrative examples are considered.

Asymptotic Behavior of Solutions of Certain Nonlinear Differential and Integro-differential Equations

Said R. Grace,Sandra Pinelas 장전수학회 2010 Proceedings of the Jangjeon mathematical society Vol.13 No.2

We establish several growth theorems for second order nonlinear differential and integro-differential equations. We also give necessary and suffcient conditions for solutions of second order non-linear differential equations to be bounded together with their frst derivatives and investigate its asymptotic behavior.

Stability Analysis of Linear Uncertain Differential Equations

Chen, Xiaowei,Gao, Jinwu Korean Institute of Industrial Engineers 2013 Industrial Engineeering & Management Systems Vol.12 No.1

Uncertainty theory is a branch of mathematics based on normolity, duality, subadditivity and product axioms. Uncertain process is a sequence of uncertain variables indexed by time. Canonical Liu process is an uncertain process with stationary and independent increments. And the increments follow normal uncertainty distributions. Uncertain differential equation is a type of differential equation driven by the canonical Liu process. Stability analysis on uncertain differential equation is to investigate the qualitative properties, which is significant both in theory and application for uncertain differential equations. This paper aims to study stability properties of linear uncertain differential equations. First, the stability concepts are introduced. And then, several sufficient and necessary conditions of stability for linear uncertain differential equations are proposed. Besides, some examples are discussed.

Stability Analysis of Linear Uncertain Differential Equations

Xiaowei Chen,Jinwu Gao 대한산업공학회 2013 Industrial Engineeering & Management Systems Vol.12 No.1

Uncertainty theory is a branch of mathematics based on normolity, duality, subadditivity and product axioms. Uncertain process is a sequence of uncertain variables indexed by time. Canonical Liu process is an uncertain process with stationary and independent increments. And the increments follow normal uncertainty distributions. Uncertain differential equation is a type of differential equation driven by the canonical Liu process. Stability analysis on uncertain differential equation is to investigate the qualitative properties, which is significant both in theory and application for uncertain differential equations. This paper aims to study stability properties of linear uncertain differential equations. First, the stability concepts are introduced. And then, several sufficient and necessary conditions of stability for linear uncertain differential equations are proposed. Besides, some examples are discussed.

미분방정식 지도에 대한 소고

박제남 ( Jae Nam Park ),장동숙 ( Dong Sook Jang ) 한국수학교육학회 2014 수학교육논문집 Vol.28 No.3

본 연구에서는 2009 개정 교육과정에 따른 수학과 교육과정에서 도입한 미분방정식 지도를 위한 수학적 모델링을 소개한다. 2014년에 1개 출판사만으로 출간된 ‘고급수학 II’의 교과서는 이계미분방정식 의 풀이를 거듭제곱 급수 방법을 사용하고 있다. 이에 따른 문제점을 알아보고 그 대안을 제시한다. 또한, 고급수학 II 교과서는 기계적 시스템을 다루고 있지만 전기적 시스템은 다루지 않고 있다. 따라서 교과서에서 다루는 일계미분방정식을 전기회로로 지도하는 방안을 제시한다. 끝으로 미분방정식 지도와 관련된 용어를 제시한다. In this paper we introduce mathematical modellings in teaching and learning differential equations which were adopted by 2009 revised curriculum. The textbook of ‘Advanced Mathematics II’ published in 2014 with one publisher includes the content of the second order differential equation by the power series method. This paper discusses the issue of the power series and gives an alternative method to explain problems of differential equation. Also, we found that the textbook of ‘Advanced Mathematics II’ used the mechanical system not electrical system in solving differential equation problems. Thus this paper suggests a method using an electric circuit in teaching and learning the first order differential equation. Finally we suggest some terminologies in the teaching and learning of differential equations.

Optimization of parameters in mathematical models of biological systems

추상목,김영희 한국전산응용수학회 2008 Journal of applied mathematics & informatics Vol.26 No.1

Under pathological stress stimuli, dynamics of a biological system can be changed by alteration of several components such as functional proteins, ultimately leading to disease state. These dynamics in disease state can be modeled using differential equations in which kinetic or system parameters can be obtained from experimental data. One of the most effective ways to restore a particular disease state of biology system (i.e., cell, organ and organism) into the normal state makes optimization of the altered components usually represented by system parameters in the differential equations. There has been no such approach as far as we know. Here we show this approach with a cardiac hypertrophy model in which we obtain the existence of the optimal parameters and construct an optimal system which can be used to find the optimal parameters. Under pathological stress stimuli, dynamics of a biological system can be changed by alteration of several components such as functional proteins, ultimately leading to disease state. These dynamics in disease state can be modeled using differential equations in which kinetic or system parameters can be obtained from experimental data. One of the most effective ways to restore a particular disease state of biology system (i.e., cell, organ and organism) into the normal state makes optimization of the altered components usually represented by system parameters in the differential equations. There has been no such approach as far as we know. Here we show this approach with a cardiac hypertrophy model in which we obtain the existence of the optimal parameters and construct an optimal system which can be used to find the optimal parameters.

NEW HOMOTOPY PERTURBATION METHOD FOR SOLVING INTEGRO-DIFFERENTIAL EQUATIONS

Kim, Kyoum Sun,Lim, Hyo Jin The Korean Society for Computational and Applied M 2012 Journal of applied mathematics & informatics Vol.30 No.5

Integro-differential equations arise in modeling various physical and engineering problems. Several numerical and analytical methods have been developed to solving such equations. We introduce the NHPM for solving nonlinear integro-differential equations. Several examples for solving integro-differential equations are presented to illustrate the efficiency of the proposed NHPM.

DIFFERENTIAL EQUATIONS ARISING FROM STIRLING POLYNOMIALS AND APPLICATIONS

김태균,김대산,장이채,권혁인,서종진 장전수학회 2016 Proceedings of the Jangjeon mathematical society Vol.19 No.2

In this paper, we study differential equations arising from Stirling polynomials and derive some new and explicit identities for Bernoulli numbers and Sirling polynomials from those differential equations.

A LOCAL-GLOBAL STEPSIZE CONTROL FOR MULTISTEP METHODS APPLIED TO SEMI-EXPLICIT INDEX 1 DIFFERENTIAL-ALGEBRAIC EUATIONS

Kulikov, G.Yu,Shindin, S.K. 한국전산응용수학회 1999 Journal of applied mathematics & informatics Vol.6 No.3

In this paper we develop a now procedure to control stepsize for linear multistep methods applied to semi-explicit index 1 differential-algebraic equations. in contrast to the standard approach the error control mechanism presented here is based on monitoring and contolling both the local and global errors of multistep formulas. As a result such methods with the local-global stepsize control solve differential-algebraic equation with any prescribed accuracy (up to round-off errors). For implicit multistep methods we give the minimum number of both full and modified Newton iterations allowing the iterative approxima-tions to be correctly used in the procedure of the local-global stepsize control. We also discuss validity of simple iterations for high accuracy solving differential-algebraic equations. Numerical tests support the the-oretical results of the paper.