A numerical analysis of a model of growth tumor
|
|
- Melinda Brittney Carter
- 5 years ago
- Views:
Transcription
1 Applied Mathematics and Computation 67 (2005) A numerical analysis of a model of growth tumor Andrés Barrea a, *, Cristina Turner b a CIEM, Universidad Nacional de Córdoba, Argentina b Universidad Nacional de Córdoba, CIEM-CONICET, FaMAF, Medina Allende sn, Cordoba 5000, Argentina Abstract In this paper we study a free boundary problem modeling the growth of tumors. The model uses the conventional ideas of nutrient diffusion and consumption by the cells. We consider the radially symmetric case of this free boundary problem. We apply a spectral numerical method to the system of equations. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Growth tumor Spectral method. Introduction and preliminaries A variety of PDE for tumor growth have been developed in the last decades. Some models are based on reaction diffusion equations [4,5]. Other models include hyperbolic equations, we refer to [ 3,6].In[3] the authors have studied a particular model with three cells populations: proliferating cells, quiescent cells and dead cells. This model includes densities P, Q, and D of proliferating, * Corresponding author. addresses: abarrea@mate.uncor.edu (A. Barrea), turner@mate.uncor.edu (C. Turner) /$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:0.06/.amc
2 346 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) quiescent and dead (necrotic) cell respectively, and concentration C of nutrients (generally oxygen). These densities satisfy a system of nonlinear first order hyperbolic equations in the tumor, with tumor surface as a free boundary. The cells in different states are assumed to be mixed within the tumor, and to have the same size. They assumed that the tumor is uniformly packed with cells, so that P þ Q þ D ¼ const N: Due to proliferation of cells and to removal of necrotic cells, there is a continuous movement of cells within the tumor and that the field velocity ~v satisfies the DarcyÕs law. They treat the tumor tissue as a porous medium, that is ~v ¼rr r pressure: In this model they assume that K Q ðcþ is increasing in C (rate of change from proliferating state to quiescent state). K P ðcþ is decreasing in C (rate of change from quiescent state to proliferating state). K D ðcþ is decreasing in C (quiescent cells become necrotic at a rate K D ðcþ). K A ðcþ is decreasing in C (the death rate by apoptosis). K B ðcþ is increasing in C (the proliferation rate). K B ðcþ > K A ðcþ. K R is a positive constant (the rate of removal of dead cells). The concentration C satisfies the next diffusion equation r 2 C kc ¼ 0enXðtÞ ðk > 0Þ C ¼ C 0 sobre oxðtþ where X is the tumor region at time t. The densities P, Q and D satisfies the following system: op ot þ divðpvþ ¼½K BðCÞ K Q ðcþ K A ðcþšp þ K P ðcþq oq ot þ divðqvþ ¼K QðCÞP ½K P ðcþþk D ðcþšq od ot þ divðdvþ ¼K AðCÞP þ K D ðcþq K R D: If we add the last equations then we obtain an equation for the pressure r Nr 2 r ¼ K B ðcþp K R D:
3 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) Now, we set c ¼ C=C 0 p ¼ P=N q ¼ Q=N we arrive at the following system of equations: r 2 c kc ¼ 0 in XðtÞ c ¼ on oxðtþ op ot þ divðprrþ ¼½K BðcÞ K Q ðcþ K A ðcþšp þ K P ðcþq in XðtÞ oq ot þ divðqrrþ ¼K QðcÞp ½K P ðcþ K D ðcþšq in XðtÞ r 2 r ¼ K R þ½k B ðcþþk R Šp þ K R q in XðtÞ where K i ðcþ ¼K i ðc 0 cþ for i ¼ A B D P Q: The pressure r on the surface of the tumor is equal to the surface tension, that is r ¼ c on oxðtþ ðc > 0Þ where is the mean curvature. The motion of the free boundary is given by the continuity equation or ~v ~n ¼ V n or o~n ¼ V n on oxðtþ where ~n is the outward normal and V n is the velocity of the free boundary of the free boundary in the outward normal direction. Given initial conditions Xð0Þ pðx 0Þ qðx 0Þ we would like to determine the family of domains X(t) and the functions p(x,t), q(x,t), c(x,t) and r(x,t) satisfying the last system. 2. The radial model and its properties We note that tumors grown in vitro are typically of spherical shape, which makes the study of radially symmetric solutions quite relevant. The radially symmetric case of the the general model is given by the following equations system: r 2 o or oc ðr2 Þ¼kc ð0 < r < RðtÞ t > 0Þ or oc ð0 tþ ¼0 cðrðtþ tþ ¼ ðt > 0Þ or
4 348 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) op op þ u ot or ¼½K BðcÞ K Q ðcþ K A ðcþšp þ K P ðcþq ½ðK B ðc þ K R Þp þ K R q K R Šp ð0 6 r 6 RðtÞ t > 0Þ oq oq þ u ot or ¼ K QðcÞp ½K P ðcþþk D ðcþšq ½ðK B ðcþþk R Þp þ K R q K R Šq ð0 6 r 6 RðtÞ t > 0Þ o r 2 or ðr2 uþ ¼½K B ðcþþk R Šp þ K R q K R uð0 tþ ¼0 ðt > 0Þ drðtþ ¼ uðrðtþ tþ ðt > 0Þ dt with initial data Rð0Þ pðr 0Þ qðr 0Þ: To transform the above free boundary problem in a problem with fixed boundary we first note that, for R(t) given cðr tþ is given by pffiffi RðtÞ sinhð k rþ r cðr tþ ¼ pffiffiffi ¼ c r sinhð k RðtÞÞ RðtÞ RðtÞ where op ot þ v op or ¼½K BðcÞ K Q ðcþ K A ðcþšp þ K P ðcþq ½ðK B ðcþþk R Þp þ K R q K R Šp ð0 6 r 6 t > 0Þ oq ot þ v oq or ¼ K QðcÞp ½K P ðcþþk D ðcþšq ½ðK B ðcþþk R Þp þ K R q K R Šq ð0 6 r 6 t > 0Þ vðr tþ ¼uðr tþ ruð tþ ð0 6 r 6 t > 0Þ o r 2 or ðr2 uþ¼½k B ðcþþk R Šp þ K R q K R ð0 6 r 6 t > 0Þ uð0 tþ ¼0 ðt > 0Þ drðtþ dt ¼ RðtÞuð tþ ðt > 0Þ
5 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) with initial data Rð0Þ ¼R 0 pðr 0Þ ¼p 0 ðrþ qðr 0Þ ¼q 0 ðrþ where R 0 > 0 p 0 ðrþ P 0 q 0 ðrþ P 0 p 0 ðrþþq 0 ðrþ 6 ð0 6 r 6 Þ: In [3] the authors show that the last system has a unique solution for 0 6 r 6, 0 6 t >, and it has the following properties:. p(r,t) P 0, q(r,t) P 0, p(r,t) +q(r,t) 6, 2. R 0 e 3 KRt 6 RðtÞ 6 R 0 e 3 KBðÞt, 3. lim t! R(t) = if K R =0, 4. d 0 6 R(t) 6 M for all t P 0if0<K R <. 3. The numerical method Spectral methods are based in simple ideas of interpolation [7], we use these ideas for the numerical method. We give a description of our numerical scheme for the problem with fixed boundary. Let r i ¼ i for 0 < i 6 N + be a partition of the interval [0,] into subintervals I i =[r i,r i+ ], of length h ¼. First, Nþ Nþ we consider the following matrix: D r i ¼ Nþ Y a i ðr i r k Þ¼ ði 6¼ Þ a a k¼ ðr i r Þ and where D r ii ¼ XNþ ðr r k Þ k¼ a ¼ YNþ ðr i r Þ: k¼ We use the following notation P ¼ðP ðtþþ ¼ðpðr tþþ Q ¼ðQ ðtþþ ¼ðqðr tþþ U ¼ðU ðtþþ ¼ðuðr tþþ V ðtþ ¼V ðtþ ¼U ðtþ U Nþ ðtþ
6 350 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) where (Æ) is a vector with components. We approximate D r P op or ðr tþ D r Q oq or ðr tþ : Now, we consider the following approximations op ot ðr tþþvðr tþ op or ðr tþ dp dt þ diagðv ÞD r P where [Æ] denote the -component of the vector (Æ). oq ot ðr tþþvðr tþ oq or ðr tþ dq dt þ diagðv ÞD r Q r 2 o or ðr2 uþðr tþ diag r 2 k D r r 2 U We remark that K i (c) =K i (r,r) for i = A,B,P,Q,D. We define the following matrices: M ðrþ ¼diagðK B ðr RÞ K Q ðr RÞ K A ðr RÞÞ M 2 ðrþ ¼diagðK P ðr RÞÞ M 3 ðrþ ¼diagðK B ðr RÞþK R Þ M 4 ðrþ ¼diagðK Q ðr RÞÞ M 5 ðrþ ¼diagðK P ðr RÞþK D ðr RÞÞ and the following vectors: P 2 ðtþ ¼ðP 2 ðtþþ Q 2 ðtþ ¼ðQ 2 ðtþþ P:QðtÞ ¼ðP ðtþq ðtþþ r 2 UðtÞ ¼ðr 2 U ðtþþ ¼ðÞ : We have the following approximations dp dt þ diagðv ÞD r P ¼ M P þ M 2 Q M 3 P 2 K R PQ þ K R P dq dt þ diagðv ÞD r Q ¼ M 4 P M 5 Q M 3 PQ K R Q 2 K R Q :
7 diag D r r 2 U ¼ M r 2 3 P þ K R Q K R k! U ¼ diag ðd r Þ diagðr 2 r 2 k ÞðM 3P þ K R Q K R Þ: From the last equation, we can obtain the vector U as function of the P, Q and R. Then we can express diag(v ) as function of P, Q, and R we obtain the following system of ordinary differential equations dp dt ¼ F ðr P QÞ dq dt ¼ F 2ðR P QÞ dr dt ¼ F 3ðR P QÞ with initial conditions A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) Pð0Þ ¼ðpðr 0ÞÞ ¼ðp 0 ðr ÞÞ Qð0Þ ¼ðqðr 0ÞÞ ¼ðq 0 ðr ÞÞ Rð0Þ ¼R 0 : If we define W ¼½P Q RŠ t then we can use the following notation dw ¼ FðWÞ dt Wð0Þ ¼W 0 : This system can be resolved by means the standard numerical methods for the ordinary differential equations. 4. Numerical examples From [6], in this section we set the following values for the parameters of the problem:. R(0) = 3, 2. k = 0.05, 3. K A (c) =0, 4. K P (c) = 0.05c, 5. K Q (c) = 0.05(c + ), 6. K D (c) = c +, 7. K B (c) =c.
8 352 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) The parameters K R and the functions p 0 (r) andq 0 (r) will be different in the examples. We set 0 points for the discretization of the interval [0, ]. A Runge Kutta method is used by integrate the ordinary differential equations. Example. In this example, we consider several values for parameter K R,we remark that lim RðtÞ ¼ if K R ¼ 0 t! we set p 0 (r) = 0.3 and q 0 (r) = 0.6. From Fig. we can see that the solution is increasing with respect to K R. Example 2. Now p 0 (r) = sin(px/2(n )) where N = 0 the number of points in [0, ] for the discretization. In Figs. 2 and 3 we show the behavior of the densities for different times. Example 3. In the last example, the parameters are the same in Example, with K R =. We remember that R 0 e 3 KRt 6 RðtÞ 6 R 0 e 3 KBðÞt Fig. 4 shows that the analytic bounds are preserved by the numerical solution. Fig.. R(t) for several values of K R.
9 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) P(r,t) t r Fig. 2. P(r,t) for different times Q(r,t) t r Fig. 3. Q(r,t) for different times.
10 354 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) R(0)exp( 0.33K R t) solution R(0)exp(0.33K B ()t) R(t) Conclusion t Fig. 4. R(t), lower and upper bounds. In this paper we have investigated the numerical solution of a model of tumor growth. An special approach of transforming a PDE system into a ODE system has been adopted, this approach is based in simple ideas of interpolation. The analytic properties of the solution are preserved for the numerical method. References [] S. Cui, A. Friedman, Analysis of a mathematical model of the effect of inhibitors on the growth of tumors, Math Biosci. 64 (2000) [2] S. Cui, A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl. 255 (200) [3] S. Cui, A. Friedman, A hyperbolic free boundary problem modeling tumor growth, Interf. Free Bound. 5 (2003) [4] J.A. Adam, A simplified mathematical model of tumor growth, Math. Biosci. 8 (986) [5] H. Byrne, M. Chaplain, Free boundary value problems associated with growth and development of multicellular spheroids, Eur. J. Appl. Math. 8 (997) [6] G. Pettet, C. Please, M. Tindall, D. McElwain, The migration of cells in multicell tumor spheroids, Bull. Math. Biol. 63 (200) [7] L. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, PA, 2000.
Universidad Nacional de Córdoba CIEM-CONICET, Argentina
Selçuk J. Appl. Math. Vol. 10. No. 1. pp. 147-155, 2009 Selçuk Journal of Applied Mathematics A Simple Discrete Model for the Growth Tumor Andrés Barrea, Cristina Turner Universidad Nacional de Córdoba
More informationExistence of positive periodic solutions for a periodic logistic equation
Applied Mathematics and Computation 139 (23) 311 321 www.elsevier.com/locate/amc Existence of positive periodic solutions for a periodic logistic equation Guihong Fan, Yongkun Li * Department of Mathematics,
More informationA new method for homoclinic solutions of ordinary differential equations
Chaos, Solitons and Fractals xxx (2007) xxx xxx www.elsevier.com/locate/chaos A new method for homoclinic solutions of ordinary differential equations F. Talay Akyildiz a, K. Vajravelu b, *, S.-J. Liao
More informationCommun Nonlinear Sci Numer Simulat
Commun Nonlinear Sci Numer Simulat 7 (0) 3499 3507 Contents lists available at SciVerse ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns Application of the
More informationChaos, Solitons and Fractals
Chaos, Solitons and Fractals 42 (2009) 3047 3052 Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos Solutions of the SIR models of epidemics
More informationChebyshev pseudospectral method for computing numerical solution of convection diffusion equation
Available online at wwwsciencedirectcom Applied Mathematics and Computation (8) 537 546 wwwelseviercom/locate/amc Chebyshev pseudospectral method for computing numerical solution of convection diffusion
More informationA three-dimensional steady-state tumor system
A three-dimensional steady-state tumor system Wenrui Hao a,1, Jonathan D. Hauenstein b,,beihu a,, Andrew J. Sommese a,1 a Department of Applied and Computational Mathematics and Statistics, University
More informationApplied Mathematical Modelling
Applied Mathematical Modelling 33 (29) 2293 2297 Contents lists available at ScienceDirect Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm Stability criteria for a nonlinear
More informationChapter 2 Linear Systems
Chapter 2 Linear Systems This chapter deals with linear systems of ordinary differential equations (ODEs, both homogeneous and nonhomogeneous equations Linear systems are etremely useful for analyzing
More informationCell cycle control and bifurcation for a free boundary problem modeling tissue growth
Cell cycle control and bifurcation for a free boundary problem modeling tissue growth Wenrui Hao Bei Hu Andrew J. Sommese April 3, 212 Abstract We consider a free boundary problem for a system of partial
More informationInput output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int J Robust Nonlinear Control 23; 13:99 937 (DOI: 112/rnc83) Input output linearization with delay cancellation for nonlinear delay systems: the problem
More informationOn the representation of hysteresis operators of Preisach type
Physica B 372 (2006) 40 44 www.elsevier.com/locate/physb On the representation of hysteresis operators of Preisach type R.V. Iyer, R. Paige Department of Mathematics and Statistics, Room 201, Broadway
More informationJournal of Computational Physics
Journal of Computational Physics 9 () 759 763 Contents lists available at ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp Short Note A comment on the computation
More informationHaar wavelet method for nonlinear integro-differential equations
Applied Mathematics and Computation 176 (6) 34 333 www.elsevier.com/locate/amc Haar wavelet method for nonlinear integro-differential equations Ülo Lepik Institute of Applied Mathematics, University of
More informationNUMERICAL METHODS FOR THE SEMICONDUCTOR BOLTZMANN EQUATION BASED ON SPHERICAL HARMONICS EXPANSIONS AND ENTROPY DISCRETIZATIONS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 TRANSPORT THEORY AND STATISTICAL PHYSICS Vol. 31, Nos. 4 6, pp. 431 452, 2002 NUMERICAL
More informationInverse-Type Sampling Procedure for Estimating the Number of Multinomial Classes
MARCEL DEKKER, INC. 27 MADISON AVENUE NEW YORK, NY 116 23 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel
More informationCommun Nonlinear Sci Numer Simulat
Commun Nonlinear Sci Numer Simulat xxx (2009) xxx xxx Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns A one-step optimal homotopy
More informationStochastic approximations of perturbed Fredholm Volterra integro-differential equation arising in mathematical neurosciences
Applied athematics and Computation 6 (27) 73 2 www.elsevier.com/locate/amc Stochastic approximations of perturbed Fredholm Volterra integro-differential equation arising in mathematical neurosciences.
More informationChapter 2 Rheological Models: Integral and Differential Representations
Chapter 2 Rheological Models: Integral and Differential Representations Viscoelastic relations may be expressed in both integral and differential forms. Integral forms are very general and appropriate
More informationAppendix B from Costinot et al., A Theory of Capital Controls as Dynamic Terms-of-Trade Manipulation (JPE, vol. 122, no. 1, p. 77)
2014 by The University of Chicago. All rights reserved. DOI: 10.1086/674425 Appendix B from Costinot et al., A Theory of Capital Controls as Dynamic Terms-of-Trade Manipulation (JPE, vol. 122, no. 1, p.
More informationSynchronization of self-sustained oscillators by common white noise
Physica A 351 (5) 16 13 www.elsevier.com/locate/physa Synchronization of self-sustained oscillators by common white noise D.S. Goldobin a,b, A.S. Pikovsky a, a Department of Physics, University of Potsdam,
More informationSome Mathematical Models of Cancer Tumors
Some Mathematical Models of Cancer Tumors by Alicia Marinis A project submitted to the Department of Mathematical Sciences in conformity with the requirements for Math 4301 (Honour s Seminar) Lakehead
More informationA new modified Halley method without second derivatives for nonlinear equation
Applied Mathematics and Computation 189 (2007) 1268 1273 www.elsevier.com/locate/amc A new modified Halley method without second derivatives for nonlinear equation Muhammad Aslam Noor *, Waseem Asghar
More informationARTICLE IN PRESS. Mechanical Systems and Signal Processing
Mechanical Systems and Signal Processing ] (]]]]) ]]] ]]] Contents lists available at ScienceDirect Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp Excitation
More informationAnalytical approximation of open-channel flow for controller design
Applied Mathematical Modelling 28 (2004) 677 695 www.elsevier.com/locate/apm Analytical approximation of open-channel flow for controller design Xavier Litrico a, *, Vincent Fromion b,1 a Cemagref, UR
More informationAnalysis of the Global Relation for the Nonlinear Schro dinger Equation on the Half-line
Letters in Mathematical Physics 65: 199 212, 23. 199 # 23 Kluwer Academic Publishers. Printed in the Netherlands. Analysis of the Global Relation for the Nonlinear Schro dinger Equation on the Half-line
More informationRounding Error in Numerical Solution of Stochastic Differential Equations
STOCHASTIC ANALYSIS AND APPLICATIONS Vol. 21, No. 2, pp. 281 300, 2003 Rounding Error in Numerical Solution of Stochastic Differential Equations Armando Arciniega* and Edward Allen Department of Mathematics
More informationInitial inverse problem in heat equation with Bessel operator
International Journal of Heat and Mass Transfer 45 (22) 2959 2965 www.elsevier.com/locate/ijhmt Initial inverse problem in heat equation with Bessel operator Khalid Masood *, Salim Messaoudi, F.D. Zaman
More informationThe distribution models of grazing animals between two grassland resource points q
Applied Mathematics and Computation xxx (004) xxx xxx www.elsevier.com/locate/amc The distribution models of grazing animals between two grassland resource points q Weiming Wang a,b, Zhenqing Li a, * a
More informationThe Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow
Journal of Computational Physics (5) www.elsevier.com/locate/jcp The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow D.W. Schwendeman *,, C.W. Wahle, A.K.
More informationPreservation of local dynamics when applying central difference methods: application to SIR model
Journal of Difference Equations and Applications, Vol., No. 4, April 2007, 40 Preservation of local dynamics when applying central difference methods application to SIR model LIH-ING W. ROEGER* and ROGER
More informationOptimal Control of a Delayed SIRS Epidemic Model with Vaccination and Treatment
Acta Biotheor DOI.7/s44-5-9244- REGULAR ARTICLE Optimal Control of a Delayed SIRS Epidemic Model with Vaccination and Treatment Hassan Laarabi Abdelhadi Abta Khalid Hattaf Received: 2 February 24 / Accepted:
More informationOptimization methods for the verification of second order sufficient conditions for bang bang controls
OPTIMAL CONTROL APPLICATIONS AND METHODS Optim. Control Appl. Meth. 25; 26:29 56 Published online April 25 in Wiley InterScience (www.interscience.wiley.com). DOI:.2/oca.756 Optimization methods for the
More informationTwo numerical methods for solving a backward heat conduction problem
pplied Mathematics and Computation 79 (6) 37 377 wwwelseviercom/locate/amc Two numerical methods for solving a backward heat conduction problem Xiang-Tuan Xiong, Chu-Li Fu *, hi Qian epartment of Mathematics,
More informationNumerical solution of a Fredholm integro-differential equation modelling _ h-neural networks
Available online at www.sciencedirect.com Applied Mathematics and Computation 195 (2008) 523 536 www.elsevier.com/locate/amc Numerical solution of a Fredholm integro-differential equation modelling _ h-neural
More informationChaos, Solitons and Fractals
Chaos, Solitons and Fractals (2008) Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos Fractal growth of tumors and other cellular populations:
More informationAn inverse problem in estimating simultaneously the effective thermal conductivity and volumetric heat capacity of biological tissue
Applied Mathematical Modelling 31 (7) 175 1797 www.elsevier.com/locate/apm An inverse problem in estimating simultaneously the effective thermal conductivity and volumetric heat capacity of biological
More informationInformation Sciences
Information Sciences 8 ) 434 457 Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins Artificial neural network approach for solving fuzzy differential
More informationHigh-order representation of Poincare maps
Nuclear Instruments and Methods in Physics Research A 558 (2006) 106 111 wwwelseviercom/locate/nima High-order representation of Poincare maps Johannes Grote, Martin Berz, Kyoko Makino Department of Physics
More informationDynamics of hierarchical models in discrete-time
Journal of Difference Equations and Applications, Vol. 11, No. 2, February 25, 95 115 Dynamics of hierarchical models in discrete-time S.R.-J. JANG * and J.M. CUSHING Department of Mathematics, University
More informationCommun Nonlinear Sci Numer Simulat
Commun Nonlinear Sci Numer Simulat 4 (9) 6 96 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns A constructive proof on the existence
More informationSynchronization of an uncertain unified chaotic system via adaptive control
Chaos, Solitons and Fractals 14 (22) 643 647 www.elsevier.com/locate/chaos Synchronization of an uncertain unified chaotic system via adaptive control Shihua Chen a, Jinhu L u b, * a School of Mathematical
More informationArbitrary order Krylov deferred correction methods for differential algebraic equations
Journal of Computational Physics 221 (2007) 739 760 www.elsevier.com/locate/jcp Arbitrary order Krylov deferred correction methods for differential algebraic equations Jingfang Huang *,1, Jun Jia 1, Michael
More informationConvergence of a linear recursive sequence
int. j. math. educ. sci. technol., 2004 vol. 35, no. 1, 51 63 Convergence of a linear recursive sequence E. G. TAY*, T. L. TOH, F. M. DONG and T. Y. LEE Mathematics and Mathematics Education, National
More informationShooting Methods for Numerical Solution of Stochastic Boundary-Value Problems
STOCHASTIC ANALYSIS AND APPLICATIONS Vol. 22, No. 5, pp. 1295 1314, 24 Shooting Methods for Numerical Solution of Stochastic Boundary-Value Problems Armando Arciniega and Edward Allen* Department of Mathematics
More informationEstimation of carbon transfer coefficients using Duke Forest free-air CO 2 enrichment data
Applied Mathematics and Computation 1 () 11 1 wwwelseviercom/locate/amc Estimation of carbon transfer coefficients using Duke Forest free-air CO enrichment data Luther White a, *, Yiqi Luo b a Department
More informationMacroscopic traffic flow propagation stability for adaptive cruise controlled vehicles
Transportation Research Part C 14 (006) 81 95 www.elsevier.com/locate/trc Macroscopic traffic flow propagation stability for adaptive cruise controlled vehicles Jingang Yi a, *, Roberto Horowitz b a Department
More informationA two level finite difference scheme for one dimensional PennesÕ bioheat equation
Applied Mathematics and Computation 171 (5) 3 331 wwwelseviercom/locate/amc A two level finite difference scheme for one dimensional PennesÕ bioheat equation Jennifer J Zhao a,1, Jun Zhang b, *,, Ning
More informationCompatible algorithms for coupled flow and transport q
Comput. Methods Appl. Mech. Engrg. 193 (24) 2565 258 www.elsevier.com/locate/cma Compatible algorithms for coupled flow and transport q Clint Dawson *, Shuyu Sun, Mary F. Wheeler Center for Subsurface
More informationA dynamic multiscale viscosity method for the spectral approximation of conservation laws
Comput. Methods Appl. Mech. Engrg. 195 (2006) 1778 1792 www.elsevier.com/locate/cma A dynamic multiscale viscosity method for the spectral approximation of conservation laws Assad A. Oberai *, John Wanderer
More informationApproximate step response of a nonlinear hydraulic mount using a simplified linear model
Journal of Sound and Vibration 99 (007) 656 663 Short Communication JOURNAL OF SOUND AND VIBRATION Approximate step response of a nonlinear hydraulic mount using a simplified linear model Song He, Rajendra
More informationModal power flow analysis of a damaged plate
Journal of Sound and Vibration 320 (2009) 84 100 JOURNAL OF SOUND AND VIBRATION www.elsevier.com/locate/jsvi Modal power flow analysis of a damaged plate.o. ong, X.Q. ang, L. Cheng Department of Mechanical
More information35-959, Rzeszów, Poland b Institute of Computer Science, Jagiellonian University,
This article was downloaded by: [Polska Akademia Nauk Instytut Matematyczny] On: 07 March 03, At: 03:7 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 0795 Registered
More informationHigher-Order Differential Equations
CHAPTER Higher-Order Differential Equations. HIGHER-ORDER HOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS. ðd 7D þ 0Þy 0 has auxiliary equation m 7m þ 00 which factors into ðm Þðm 5Þ 0 and so has roots
More informationAxial Flow Compressors and Fans
5 Axial Flow Compressors and Fans 5.1 INTRODUCTION As mentioned in Chapter 4, the maximum pressure ratio achieved in centrifugal compressors is about 4:1 for simple machines (unless multi-staging is used)
More informationEfficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods
Journal of Computational Physics 213 (2006) 748 776 wwwelseviercom/locate/jcp Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods M Tokman * Department
More informationThis article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author s institution, sharing
More informationJournal of Computational Physics
Journal of Computational Physics 8 (009) 8 93 Contents lists available at ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp A mixed implicit explicit finite difference
More informationA direct Eulerian GRP scheme for compressible fluid flows
Journal of Computational Physics 8 (6) 9 43 www.elsevier.com/locate/jcp A direct Eulerian GP scheme for compressible fluid flows Matania Ben-Artzi a, Jiequan i b, *, Gerald Warnecke c a Department of Mathematics,
More informationA homotopy method for determining the eigenvalues of locally or non-locally reacting acoustic liners in flow ducts
Journal of Sound and Vibration 303 (2007) 277 286 JOURNAL OF SOUND AND VIBRATION www.elsevier.com/locate/jsvi A homotopy method for determining the eigenvalues of locally or non-locally reacting acoustic
More informationChapter 2 Principles of Analytical Mechanics
Chapter 2 Principles of Analytical Mechanics 2.1 Constraints As we have already mentioned in Chap. 1, a particle (or a system of particles) is subject to constraints if its motion is restricted by a constraint
More informationA moving grid finite element method applied to a model biological pattern generator q
Journal of Computational Physics 190 (2003) 478 500 www.elsevier.com/locate/jcp A moving grid finite element method applied to a model biological pattern generator q Anotida Madzvamuse a, *, Andrew J.
More informationCONTINUOUS-TIME Markov chains (CTMSs) are an established
IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 37, NO. X, XXXXXXX 2011 1 Scalable Differential Analysis of Process Algebra Models Mirco Tribastone, Stephen Gilmore, and Jane Hillston Abstract The exact
More informationStability of macroscopic traffic flow modeling through wavefront expansion
Transportation Research Part B 37 (2003) 661 679 www.elsevier.com/locate/trb Stability of macroscopic traffic flow modeling through wavefront expansion Jingang Yi a,1, Hao Lin a,2, Luis Alvarez b, Roberto
More informationHow do our representations change if we select another basis?
CHAPTER 6 Linear Mappings and Matrices 99 THEOREM 6.: For any linear operators F; G AðV Þ, 6. Change of Basis mðg FÞ ¼ mðgþmðfþ or ½G FŠ ¼ ½GŠ½FŠ (Here G F denotes the composition of the maps G and F.)
More informationLinearization techniques for singular initial-value problems of ordinary differential equations
Applied Mathematics and Computation 161 (25) 52542 www.elsevier.com/locate/amc Linearization techniques for singular initial-value problems of ordinary differential equations J.I. Ramos Room I-32-D, E.T.S.
More informationCommun Nonlinear Sci Numer Simulat
Commun Nonlinear Sci Numer Simulat 16 (2011) 2730 2736 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns Homotopy analysis method
More informationAn inverse vibration analysis of a tower subjected to wind drags on a shaking ground
Applied Mathematical Modelling 26 (2002) 517 528 www.elsevier.com/locate/apm An inverse vibration analysis of a tower subjected to wind drags on a shaking ground C.C. Kang a, C.Y. Lo b, * a Department
More informationARTICLE IN PRESS. Physica E
Physica E 41 (2009) 1651 1655 Contents lists available at ScienceDirect Physica E journal homepage: www.elsevier.com/locate/physe A general nonlocal beam theory: Its application to nanobeam bending, buckling
More informationGold nanoparticle ensembles as heaters and actuators: melting and collective plasmon resonances
Nanoscale Res Lett (6) 1:8 9 DOI 1.17/s11671-6-915-7 NANO EXPRESS Gold nanoparticle ensembles as heaters and actuators: melting and collective plasmon resonances Alexander O. Govorov Æ Wei hang Æ Timur
More informationExperimental designs for precise parameter estimation for non-linear models
Minerals Engineering 17 (2004) 431 436 This article is also available online at: www.elsevier.com/locate/mineng Experimental designs for precise parameter estimation for non-linear models Z. Xiao a, *,
More informationCommun Nonlinear Sci Numer Simulat
Commun Nonlinear Sci Numer Simulat 14 (2009) 3833 3843 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns Homotopy analysis solution
More informationApplied Mathematical Modelling
Applied Mathematical Modelling () Contents lists available at SciVerse ScienceDirect Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm Adaptive non-fragile observer design for
More informationNumerical solution of hyperbolic heat conduction in thin surface layers
International Journal of Heat and Mass Transfer 50 (007) 9 www.elsevier.com/locate/ijhmt Numerical solution of hyperbolic heat conduction in thin surface layers Tzer-Ming Chen * Department of Vehicle Engineering,
More informationQualitative analysis of a chemostat model with inhibitory exponential substrate uptake q
Chaos, Solitons and Fractals 23 (2005) 873 886 www.elsevier.com/locate/chaos Qualitative analysis of a chemostat model with inhibitory exponential substrate uptake q Guifang Fu a, *, Wanbiao Ma a, Shigui
More informationJournal of Computational Physics
Journal of Computational Physics 229 (2010 7747 7764 Contents lists available at ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp A unified gas-kinetic scheme
More informationA discontinuous Galerkin finite element method for directly solving the Hamilton Jacobi equations
Journal of Computational Physics 3 (7) 39 415 www.elsevier.com/locate/jcp A discontinuous Galerkin finite element method for directly solving the Hamilton Jacobi equations Yingda Cheng, Chi-Wang Shu *
More informationHigh-order well-balanced schemes and applications to non-equilibrium flow
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln NASA Publications National Aeronautics and Space Administration 29 High-order well-balanced schemes and applications to
More informationElectronic energy transfer in polymers labeled at both ends with fluorescent groups
Journal of Luminescence 96 (22) 269 278 Electronic energy transfer in polymers labeled at both ends with fluorescent groups E.. Bodunov, M.. Berberan-Santos, J.M.G. Martinho* Centro de Qu!ımica-F!ısica
More informationCommun Nonlinear Sci Numer Simulat
Commun Nonlinear Sci Numer Simulat 15 (21) 3965 3973 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns A similarity solution in
More informationDynamic scaling in a simple one-dimensional model of dislocation activity
Philosophical Magazine, 11 August 24 Vol. 84, No. 23, 2445 2454 Dynamic scaling in a simple one-dimensional model of dislocation activity Jack Deslippey, R. Tedstromy, Murray S. Daw*y, D. Chrzanz}, T.
More informationEquivalence between transfer-matrix and observed-state feedback control
Equivalence between transfer-matrix and observed-state feedback control C.R. Rojas, R.A. Rojas and M.E. Salgado Abstract: An observed-state feedback is built for a given multiple input multiple output
More informationPRIVACY is becoming an increasingly important issue in
92 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 18, NO. 1, JANUARY 2006 Random Projection-Based Multiplicative Data Perturbation for Privacy Preserving Distributed Data Mining Kun Liu, Hillol
More informationCounting Singular Matrices with Primitive Row Vectors
Monatsh. Math. 44, 7 84 (005) DOI 0.007/s00605-004-050-7 Counting Singular Matrices with Primitive Row Vectors By Igor Wigman Tel Aviv University, Israel Communicated by W. M. Schmidt Received March 3,
More informationJournal of Process Control
Journal of Process Control 9 (29) 539 549 Contents lists available at ScienceDirect Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont Application of reduced models for robust
More informationHamiltonian formulation: water waves Lecture 3
Hamiltonian formulation: water waves Lecture 3 Wm.. Hamilton V.E. Zakharov Hamiltonian formulation: water waves This lecture: A. apid review of Hamiltonian machinery (see also extra notes) B. Hamiltonian
More informationNewton-homotopy analysis method for nonlinear equations
Applied Mathematics and Computation 188 (2007) 1794 1800 www.elsevier.com/locate/amc Newton-homotopy analysis method for nonlinear equations S. Abbasbandy a, *, Y. Tan b, S.J. Liao b a Department of Mathematics,
More informationMinimal almost convexity
J. Group Theory 8 (2005), 239 266 Journal of Group Theory ( de Gruyter 2005 Minimal almost convexity Murray Elder and Susan Hermiller* (Communicated by D. J. S. Robinson) Abstract. In this article we show
More informationEllipsoidally-shaped local absorbing boundaries for three-dimensional scalar wave propagation
Comput Mech (004) DOI 0.007/s00466-004-0585-x ORIGINAL PAPER L. F. Kallivokas Æ S. Lee Ellipsoidally-shaped local absorbing boundaries for three-dimensional scalar wave propagation Received: September
More informationGeneralized diffusion equation
Physica A 368 (006) 55 6 www.elsevier.com/locate/hysa Generalized diffusion equation Jean Pierre Boon, James F. Lutsko Center for Nonlinear Phenomena and Comlex Systems, Université Libre de Bruxelles,
More informationApplied Mathematics and Computation
Applie Mtemtics n Computtion 215 (2010) 3713 3720 Contents lists vilble t ScienceDirect Applie Mtemtics n Computtion journl omepge www.elsevier.com/locte/mc Positive solutions to semi-positone secon-orer
More informationLimitations on the utility of exact master equations
Annals of Physics 39 (005) 348 363 www.elsevier.com/locate/aop Limitations on the utility of exact master equations G.W. Ford, R.F. OÕConnell *, School of Theoretical Physics, Dublin Institute for Advanced
More informationFully-coupled nonlinear analysis of single lap adhesive joints
International Journal of Solids and Structures (7) 9 7 www.elsevier.com/locate/ijsolstr Fully-coupled nonlinear analysis of single lap adhesive joints Quantian Luo, Liyong Tong * School of Aerospace, Mechanical
More informationWilliamsburg, VA, USA b Harvard School of Engineering and Applied Sciences, Cambridge, University, Norfolk, VA, USA
This article was downloaded by: [Jianjun Paul Tian] On: 19 July 212, At: :27 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 172954 Registered office: Mortimer
More informationCommun Nonlinear Sci Numer Simulat
Commun Nonlinear Sci Numer Simulat 18 (2013) 878 887 Contents lists available at SciVerse ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns A continuous/discrete
More informationAn Explicit Formula for the Discrete Power Function Associated with Circle Patterns of Schramm Type
Funkcialaj Ekvacioj, 57 (2014) 1 41 An Explicit Formula for the Discrete Power Function Associated with Circle Patterns of Schramm Type By Hisashi Ando1, Mike Hay2, Kenji Kajiwara1 and Tetsu Masuda3 (Kyushu
More informationChapter 2 Torsion Stresses in Thin-Walled Multi-Cell Box-Girders
Chapter Torsion Stresses in Thin-Walled Multi-Cell Box-Girders. Torsion of Uniform Thin-Walled Two-Cell Box-Girders The thin-walled box section with uniform thickness t as shown in Fig.., is subjected
More informationTHE ALGORITHM TO CALCULATE THE PERIOD MATRIX OF THE CURVE x m þ y n ¼ 1
TSUKUA J MATH Vol 26 No (22), 5 37 THE ALGORITHM TO ALULATE THE PERIOD MATRIX OF THE URVE x m þ y n ¼ y Abstract We show how to take a canonical homology basis and a basis of the space of holomorphic -forms
More informationJournal of Computational Physics
ournal of Computational Physics 9 () 397 3987 Contents lists available at ScienceDirect ournal of Computational Physics journal homepage: www.elsevier.com/locate/jcp A multiscale coupling method for the
More informationRKC time-stepping for advection diffusion reaction problems
Journal of Computational Physics 21 (24) 61 79 www.elsevier.com/locate/jcp RKC time-stepping for advection diffusion reaction problems J.G. Verwer *, B.P. Sommeijer, W. Hundsdorfer CWI, P.O. Box 9479,
More information