(Luận án tiến sĩ) phát triển phương pháp sai phân khác thường giải một số lớp phương trình vi phân

MINISTRY OF EDUCATION AND VIETNAM ACADEMY TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCES AND TECHNOLOGY Hoàng Mạnh Tuấn DEVELOPMENT OF NONSTANDARD FINITE DIFFERENCE METHODS FOR SOME CLASSES OF DIFFERENTIAL EQUATIONS DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI - 2021 MINISTRY OF EDUCATION AND VIETNAM ACADEMY TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCES AND TECHNOLOGY Hoàng Mạnh Tuấn DEVELOPMENT OF NONSTANDARD FINITE DIFFERENCE METHODS FOR SOME CLASSES OF DIFFERENTIAL EQUATIONS Speciality: Applied Mathematics Speciality Code: 46 01 12 DOCTOR OF PHILOSOPHY IN MATHEMATICS SUPERVISORS: Prof Dr Đặng Quang Á Assoc Prof Dr Habil Vũ Hoàng Linh HANOI - 2021 BỘ GIÁO DỤC VÀ ĐÀO TẠO VIỆN HÀN LÂM KHOA HỌC VÀ CÔNG NGHỆ VIỆT NAM HỌC VIỆN KHOA HỌC VÀ CƠNG NGHỆ Hồng Mạnh Tuấn PHÁT TRIỂN PHƯƠNG PHÁP SAI PHÂN KHÁC THƯỜNG GIẢI MỘT SỐ LỚP PHƯƠNG TRÌNH VI PHÂN LUẬN ÁN TIẾN SĨ TOÁN HỌC HÀ NỘI - 2021 BỘ GIÁO DỤC VÀ ĐÀO TẠO VIỆN HÀN LÂM KHOA HỌC VÀ CÔNG NGHỆ VIỆT NAM HỌC VIỆN KHOA HỌC VÀ CƠNG NGHỆ Hồng Mạnh Tuấn PHÁT TRIỂN PHƯƠNG PHÁP SAI PHÂN KHÁC THƯỜNG GIẢI MỘT SỐ LỚP PHƯƠNG TRÌNH VI PHÂN Chuyên ngành: Toán ứng dụng Mã số: 46 01 12 LUẬN ÁN TIẾN SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC: GS TS Đặng Quang Á PGS TSKH Vũ Hoàng Linh HÀ NỘI - 2021 Lời cam đoan Luận án hoàn thành Học viện Khoa học Công nghệ, Viện Hàn lâm Khoa học công nghệ Việt Nam hướng dẫn khoa học GS TS Đặng Quang Á PGS TSKH Vũ Hoàng Linh Những kết nghiên cứu trình bày luận án mới, trung thực chưa cơng bố cơng trình khác Các kết công bố chung cán hướng dẫn cho phép sử dụng luận án Hà Nội, tháng 01 năm 2021 Nghiên cứu sinh Hoàng Mạnh Tuấn i Declaration This thesis has been completed at Graduate University of Science and Technology (GUST), Vietnam Academy of Science and Technology (VAST) under the supervision of Prof Dr Đặng Quang Á and Assoc Prof Dr Habil Vũ Hoàng Linh I hereby declare that all the results presented in this thesis are new, original and have never been published fully or partially in any other work The author Hoàng Mạnh Tuấn ii Lời cảm ơn Trước hết, xin bày tỏ lòng biết ơn chân thành sâu sắc tới cán hướng dẫn, GS TS Đặng Quang Á GS TSKH Vũ Hoàng Linh Luận án khơng thể hồn thành khơng có hướng dẫn giúp đỡ tận tình Thầy Tôi vô biết ơn giúp đỡ mà Thầy dành cho không thời gian thực luận án mà suốt thời gian học Đại học Cao học Sự quan tâm giúp đỡ Thầy công việc lẫn sống giúp vượt qua những khó khăn thất vọng để hồn thiện cơng trình nghiên cứu hồn thành luận án Tơi xin gửi lời cảm ơn tới Học viện Khoa học Công nghệ, Viện Hàn lâm Khoa học Công nghệ Việt Nam, nơi học tập, nghiên cứu hoàn thành luận án Luận án hoàn thành cách thuận lợi thời hạn nhờ vào công tác quản lý đào tạo chuyên nghiệp, môi trường học tập nghiên cứu khoa học lý tưởng với giúp đỡ nhiệt tình cán Học viện Tôi xin chân thành cảm ơn Lãnh đạo đồng nghiệp Viện Công nghệ Thông tin, Viện Hàn lâm Khoa học Công nghệ Việt Nam, nơi tơi cơng tác, dàng điều kiện thuận lợi cho suốt nhiều năm qua nói chung thời gian thực luận án nói riêng Tơi xin gửi cảm ơn tới Thầy Cô, anh chị bạn bè đồng nghiệp Seminar "Toán ứng dụng" GS Đặng Quang Á chủ trì, đặc biệt cá nhân TS Nguyễn Cơng Điều, ý kiến sâu sắc, có chất lượng cao mặt học thuật buổi trao đổi chun mơn Những điều giúp tơi hồn thiện tốt cơng trình nghiên cứu Tơi xin chân thành cảm ơn các anh, chị đồng nghiệp Bộ mơn Tốn học, trường ĐH FPT, giúp đỡ động viên suốt trình thực luận án Điều tạo cho tơi nhiều cảm hứng nghiên cứu khoa học thực luận án Đặc biệt, Tôi xin gửi lời biết ơn sâu sắc tới GS TSKH Phạm Kỳ Anh, người Thầy giảng dạy hướng dẫn tận tình tơi suốt thời gian học Đại học Cao học Những giảng thầy mơn học Giải tích số Tốn ứng dụng từ thời Đại học có ảnh hưởng to lớn tới lựa chọn sau đường iii nghiên cứu khoa học Đặc biệt, Thầy có nhiều góp ý sâu sắc quan trọng giúp cho luận án hoàn thiện tốt Tôi xin gửi lời cảm ơn chân thành tới GS R E Mickens (Clark Atlanta University), GS M Ehrhardt (Bergische Universitat Wuppertal), GS A J Arenas (Universidad de Córdoba), GS J Cresson (Université de Pau et des Pays de l’Adour) nhiều đồng nghiệp nước khác dành nhiều thời gian đọc cho tơi nhiều ý kiến giá trị nội dung lẫn hình thức trình bày luận án Tơi xin chân thành cảm nhiều Giáo sư, Thầy Cô nhiều bạn bè đồng nghiệp khác dành nhiều thời gian đọc cho nhiều ý kiến giá trị hình thức trình bày luận án Tơi xin gửi lời cảm ơn chân thành tới Ths Đặng Quang Long (Viện CNTT) góp ý giá trị quan trọng cho nội dung hình thức trình bày luận án Tôi xin gửi lời cảm ơn tới tất bạn bè đồng nghiệp, người dành cho nhiều quan tâm động viên sống lẫn nghiên cứu khoa học Cuối cùng, luận án khơng thể hồn thành khơng có giúp đỡ, động viên khích lệ mặt gia đình Tơi khơng thể diễn đạt hết lời biết ơn gia đình Với tất lịng biết ơn sâu sắc, luận án nói riêng tất điều tốt đẹp mà cố gắng thực để gửi tới Bố Mẹ, vợ con, anh, chị, em người thân gia đình, người với yêu thương, đức kiên nhẫn lịng vị tha khích lệ động viên theo đuổi đường nghiên cứu khoa học suốt năm qua Hà Nội, tháng 01 năm 2021 Nghiên cứu sinh Hoàng Mạnh Tuấn iv Acknowledgments Firstly, I would like to thank my two supervisors Prof Dr Habil Vũ Hoàng Linh and especially Prof Dr Đặng Quang Á for the continuous support of my PhD study and related research; for their patience, motivation and immense knowledge Without their help I could not have overcome the difficulties in research and study The wonderful research environment of the Graduate University of Sciences and Technology, Vietnam Academy of Science and Technology, and the excellence of its staff have helped me to complete this work within the schedule I would like to thank all the staff at the Graduate University of Sciences and Technology for their help and support during the years of my PhD studies I would like to thank my big family for their endless love and unconditional support Last but not least, I would like to thank my colleagues and many other people beside me for their love, motivation and constant guidance Thanks all for your encouragement! The author Hoàng Mạnh Tuấn v List of notations and abbreviations N The set of natural numbers N+ The set of non-negative nature numbers R The set of real numbers R+ The set of non-negative real numbers Rn Real coordinate space of n-dimension Rn+ The set of all the n-tuples with non-negative real numbers σ(A) The set of the eigenvalues of the matrix A |z| The modulus of the complex number z x The norm of the vector x y(t), ˙ y (t), dy(t)/dt The first derivative of the function y(t) DDE Delay differential equation EEFD Explicit exact finite difference EFD Exact finite difference ENRK Explicit nonstandard Runge-Kutta ESRK Explicit standard Runge-Kutta FD Finite difference FDE Fractional differential equation GAS Global asymptotic stability/Globally asymptotically stable IEFD Implicit exact finite difference IVP Initial value problem HBV Hepatitis B virus LAS Local asymptotic stability/Locally asymptotically stable NSFD Nonstandard finite difference ODE Ordinary differential equation PDE Partial differential equation RK2 The second order Runge-Kutta method RK4 The classical four stage Runge-Kutta method SFD Standard finite difference w.r.t with respect to T r(J) The trace of the matrix J vi GENERAL CONCLUSIONS In this thesis, we have successfully developed the Mickens’ methodology to construct nonstandard finite difference (NSFD) methods for solving some important classes of differential equations arising in fields of science and technology The proposed NSFD schemes are not only dynamically consistent with the differential equation models, but also easy to be implemented; furthermore, they can be used to solve a large class of mathematical problems in both theory and practice The validity of the theoretical results and the superiority of the NSFD schemes have been confirmed and supported by many numerical simulations The results have indicated that there is a good agreement between the theoretical aspect and experimental one In the first part, we have successfully constructed NSFD schemes for some mathematical models described by systems of ODEs including two metapopulation models, one predator-prey model and two computer virus propagation models It is worth noting that all of the models possess at least one of the following characteristics: (i) having large dimensions (ii) having non-hyberbolic equilibrium points (iii) having the GAS Firstly, we have investigated the GAS of the constructed NSFD schemes for the metapopulation model formulated in [97] by using the standard techniques of mathematical analysis Secondly, we have used the Lyapunov stability theorem to study the GAS of proposed NSFD schemes for the computer virus propagation model and the general predator-prey model constructed in [12] and [105], respectively Lastly, we have proposed two novel approaches to establish the stability properties of the NSFD schemes for the metapopulation model and the propagation model of computer viruses formulated in [98] and [16], respectively The first approach is based on the extension of the classical Lyapunov stability theorem, and the second one is based on the Lyapunov stability theorem and its extensions in combination with a theorem on the GAS of discrete-time nonlinear cascade systems Both approaches lead the study of the stability of the proposed NSFD schemes to the study of the stability of 158 discrete models with smaller dimension, and therefore, complicated calculations and transforms are limited significantly In the second part, we have constructed EFD schemes and high order NSFD schemes for a class of general dynamical systems based on the general Runge-Kutta methods The obtained results can be considered as a generalization of the results formulated in [28, 29, 42] Firstly, implicit and explicit EDS schemes for systems of three linear ODEs with constant coefficients are constructed Importantly, the obtained results not only answer the open question posted by Roeger [40] but also can be extended to design EFD schemes for general n-dimensional systems of linear ODEs with constant coefficients Next, we have constructed and analyzed high order ENRK methods preserving two important properties of general autonomous dynamical systems, namely, the positivity and LAS The main result resolved the contradiction between the dynamics consistency and high order of accuracy of NSFD schemes Additionally, two important applications of the constructed ENRK methods to the predator-prey model and the vaccination model are also presented In the near future, the established results in this thesis will be developed to construct highly effective NSFD schemes for PDEs, DDEs, FDEs and stochastic differential equations Also, we intent to study the combination of the Mickens’ methodology and other existing approaches to create new numerical methods with high performance for both differential equations and integro-differential equations 159 THE LIST OF THE WORKS OF THE AUTHOR RELATED TO THE THESIS [A1] Quang A Dang, Manh Tuan Hoang, Dynamically consistent discrete metapopulation model, Journal of Difference Equations and Applications, 2016, 22, 1325-1349, (SCI-E) [A2] Quang A Dang, Manh Tuan Hoang, Lyapunov direct method for investigating stability of nonstandard finite difference schemes for metapopulation models, Journal of Difference Equations and Applications, 2018, 24, 15-47, (SCI-E) [A3] Quang A Dang, Manh Tuan Hoang, Complete global stability of a metapopulation model and its dynamically consistent discrete models, Qualitative Theory of Dynamical Systems, 2019, 18, 461-475, (SCI-E) [A4] Quang A Dang, Manh Tuan Hoang, Numerical dynamics of nonstandard finite difference schemes for a computer virus propagation model, International Journal of Dynamics and Control, 2020, 8, 772-778, (SCOPUS) [A5] Quang A Dang, Manh Tuan Hoang, Nonstandard finite difference schemes for a general predator-prey system, Journal of Computational Science, 2019, 36, 101015, (SCI-E) [A6] Quang A Dang, Manh Tuan Hoang, Positivity and global stability preserving NSFD schemes for a mixing propagation model of computer viruses, Journal of Computational and Applied Mathematics 2020, 374, 112753, (SCI) [A7] Manh Tuan Hoang, On the global asymptotic stability of a predator-prey model with Crowley-Martin function and stage structure for prey, Journal of Applied Mathematics and Computing, 2020, 64, 765-780, (SCI-E) [A8] Quang A Dang, Manh Tuan Hoang, Exact finite difference schemes for three-dimensional linear systems with constant coefficients, Vietnam Journal of Mathematics, 2018, 46, 471-492, (ESCI, SCOPUS) 160 [A9] Quang A Dang, Manh Tuan Hoang, Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems, International Journal of Computer Mathematics, 2020, 97, 2036-2054, (SCI-E) 161 Bibliography R P Agarwal, An Introduction to Ordinary Differential Equations, Springer, 2000 L.J.S Allen, An Introduction to Mathematical Biology, Prentice Hall, 2007, New Jersey U M Ascher, L.R Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Society for Industrial and Applied Mathematics, 1998, Philadelphia F Brauer, C Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2001, New York L Edelstein-Keshet, Mathematical Models in Biology, Society for Industrial and Applied Mathematics, 1998, Philadelphia H K Khalil, Nonlinear systems, Prentice Hall, 2002 M Martcheva, An Introduction to Mathematical Epidemiology, Springer Science+Business Media New York, 2015 R.Mattheij, J Molenaar, Classics in Applied Mathematics: Ordinary Differential Equations in Theory and Practice, Society for Industrial and Applied Mathematics, 2002, New York L Perko, Differential Equations and Dynamical Systems, Third Edition, Texts in Applied Mathematics 7, Springer, 2001 10 H L Smith, P Waltman, The Theory of the Chemostat, Cambridge University Press, 1995, Cambridge 11 W H Murray, The application of epidemiology to computer viruses, Computers & Security, 1988, 7, 130-50 12 L X Yang, X Yang, Q Zhu, L Wen, A computer virus model with graded cure rates, Nonlinear Analysis: Real World Applications, 2013, 14, 414 -422 162 13 L-X Yang, X Yang, A new epidemic model of computer viruses, Communications in Nonlinear Science and Numerical Simulation, 2014, 19, 1935-1944 14 X Yang, B K Mishra, Y Liu, Computer virus: theory, model, and methods, Discrete Dynamics in Nature and Society, 2012, Article ID 473508 15 L X Yang, X Yang, L Wen, J Liu, A novel computer virus propagation model and its dynamics, International Journal of Computer Mathematics, 2012, 89, 2307-2314 16 Q Zhu, X Yang, L Yang, X Zhang, A mixing propagation model of computer viruses and countermeasures, Nonlinear Dynamics, 2013, 73, 1433-1441 17 A M Stuart, A R Humphries, Dynamical Systems and Numerical Analysis, Cambridge monographs on applied and computational mathematics 2, Cambridge University Pres, 1998, New York 18 R L Burden, J D Faires, Numerical analysis, Brooks/Cole, Cengage Learning, 2011 19 E Hairer, P S Norsett, G Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Series in Computational Mathematics 8, SpringerVerlag Berlin Heidelberg, 1993 20 E Hairer, G Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics 14, Springer-Verlag Berlin Heidelberg, 1996 21 R E Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, 1994 22 R E Mickens, Applications of Nonstandard Finite Difference Schemes, World Scientific, 2000 23 R E Mickens, Advances in the Applications of Nonstandard Finite Difference Schemes, World Scientific, 2005 163 24 R E Mickens, Nonstandard Finite Difference Schemes for Differential Equations, Journal of Difference Equations and Applications, 2012, 8, 823-847 25 R Anguelov, J M S Lubuma, Nonstandard finite difference method by nonlocal approximations, Mathematics and Computers in Simulation, 2003, 61, 465-475 26 J Cresson, F Pierret, Non standard finite difference scheme preserving dynamical properties, Journal of Computational and Applied Mathematics, 2016, 303, 15-30 27 D T Wood, H V Kojouharov, A class of nonstandard numerical methods for autonomous dynamical systems, Applied Mathematics Letters, 2015, 50, 78-82 28 D T Dimitrov, H V Kojouharov, Nonstandard finite-difference schemes for general two-dimensional autonomous dynamical systems, Applied Mathematics Letters, 2005, 18, 769-774 29 D T Dimitrov, H V Kojouharov, Stability-preserving finite-difference methods for general multi-dimensional autonomous dynamical systems, International journal of numerical analysis and modeling, 2007, 4, 280-290 30 P Liu and S N Elaydi, Discrete competitive and cooperative models of LotkaVolterra type, Journal of Computational Analysis and Applications, 2001, 3, 53-73 31 R E Mickens, K Oyedeji, S Rucker, Exact finite difference scheme for secondorder, linear ODEs having constant coefficients, Journal of Sound and Vibration, 2005, 287, 1052-1056 32 R E Mickens, Exact finite difference schemes for two-dimensional advection equations, Journal of Sound and Vibration, 1997, 207, 426-428 33 L.-I W Roeger, R E Mickens, Exact finite-difference schemes for first order differential equations having three distinct fixed-points, Journal of Difference Equations and Applications, 2007, 13, 1179-1185 34 L -I W Roeger, R E Mickens, Exact finite difference and non-standard finite difference schemes for dx/dt = −λy α , Journal of Difference Equations and Applications, 2012, 18, 1511-1517 164 35 L -I W Roeger, Exact nonstandard finite-difference methods for a linear systemthe case of centers, Journal of Difference Equations and Applications, 2008, 14, 381-389 36 L -I W Roeger, R E Mickens, Exact finite difference scheme for linear differential equation with constant coefficients, Journal of Difference Equations and Applications, 2013, 19, 1663-1670 37 L -I W Roeger, Exact finite-difference schemes for two-dimensional linear systems with constant coefficients, Journal of Computational and Applied Mathematics, 2008, 219, 102-109 38 K C Patidar, On the use of nonstandard finite difference methods, Journal of Difference Equations and Applications, 2005, 11,11 735-758 39 K C Patidar, Nonstandard finite difference methods: recent trends and further developments, Journal of Difference Equations and Applications, 2016, 22, 817849 40 L.-I W Roeger, General nonstandard finite-difference schemes for differential equations with three fixed-points, Computers and Mathematics with Applications, 2009, 57, 379-383 41 L -I W Roeger, Nonstandard finite difference schemes for differential equations with n + distinct fixed-points, Journal of Difference Equations and Applications, 2009, 15, 133-151 42 L -I W Roeger, R E Mickens, Non-standard finite difference schemes for the differential equation y (t) = bn y n +bn−1 y n−1 + .+b1 y+b0 , Journal of Difference Equations and Applications, 2002, 18, 305-312 43 B M Chen-Charpentier, D T Dimitrov, H V Kojouharov, Combined nonstandard numerical methods for ODEs with polynomial right-hand sides, Mathematics and Computers in Simulation, 2006, 73, 105-113 165 44 F J Solis, B Chen-Charpentier, Nonstandard discrete approximations preserving stability properties of continuous mathematical models, Mathematical and Computer Modelling, 2004, 40, 481-490 45 R E Mickens, Discretizations of nonlinear differential equations using explicit nonstandard methods, Journal of Computational and Applied Mathematics, 1999, 110, 1810-185 46 R E Mickens, I Ramadhani, Finite-difference schemes having the correct linear stability properties for all finite step-sizes III, Computers & Mathematics with Applications, 1994, 27, 77-84 47 U Erdogan, T Ozis, A smart nonstandard finite difference scheme for second order nonlinear boundary value problems, Journal of Computational Physics, 2011, 230, 6464-6474 48 H Kojouharov, B Welfert, A nonstandard Euler scheme for y +g(y)y +f (y) = 0, Journal of Computational and Applied Mathematics, 2003, 151, 335-353 49 P A Zegeling, S Iqbal, Nonstandard finite differences for a truncated BratuPicard model, Applied Mathematics and Computation, 2018, 324, 266-284 50 R E Mickens, A nonstandard finite-difference scheme for the Lotka-Volterra system, Applied Numerical Mathematics, 2003, 45, 309-314 51 L -I W Roeger, A nonstandard discretization method for Lotka-Volterra models that preserves periodic solutions, Journal of Difference Equations and Applications, 2005, 11, 721-733 52 L -I W Roeger, Periodic solutions preserved by nonstandard finite-difference schemes for the Lotka–Volterra system: a different approach, Journal of Difference Equations and Applications, 2008, 14, 481-493 53 L -I W Roeger, Nonstandard finite-difference schemes for the Lotka-Volterra systems: generalization of Mickens’s method, Journal of Difference Equations and Applications, 2006, 12, 937-948 166 54 L -I W Roeger, Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite-difference schemes, Discrete & Continuous Dynamical Systems B, 2008, 9, 415-429 55 L -I W Roeger, G L Jr, Dynamically consistent discrete Lotka-Volterra competition systems, Journal of Difference Equations and Applications, 2003, 19, 191-200 56 D T Dimitrov, H V Kojouharov, Positive and elementary stable nonstandard numerical methods with applications to predator-prey models, Journal of Computational and Applied Mathematics, 2006, 189, 98-108 57 D T Dimitrov, H V Kojouharov, Nonstandard finite-difference methods for predator-prey models with general functional response, Mathematics and Computers in Simulation, 2008, 78, 1-11 58 R Anguelov, Y Dumont, J M S Lubuma, M Shillor, Dynamically consistent nonstandard finite difference schemes for epidemiological models, Journal of Computational and Applied Mathematics, 2014, 255, 161-182 59 A J Arenas, G González-Parra, B M Chen-Charpentier, A nonstandard numerical scheme of predictor-corrector type for epidemic models, Computers and Mathematics with Applications, 2010, 59, 3740-3749 60 O F Egbelowo, Nonstandard finite difference approach for solving 3-compartment pharmacokinetic models, International Journal for Numerical Methods in Biomedical Engineering, 2018, https://doi.org/10.1002/cnm.3114 61 S.M.Garba, A.B.Gumel, J M S Lubuma, Dynamically-consistent non-standard finite difference method for an epidemic model, Mathematical and Computer Modelling, 2011, 53, 131-150 62 K F Gurski, A simple construction of nonstandard finite-difference schemes for small nonlinear systems applied to SIR models, Computers and Mathematics with Applications, 2013, 66, 2165-2177 167 63 L Jodar, R J.Villanueva, A J Arenas, G C.González, Nonstandard numerical methods for a mathematical model for influenza disease, Mathematics and Computers in Simulation, 2008, 79, 622-633 64 A Korpusik, A nonstandard finite difference scheme for a basic model of cellular immune response to viral infection, Communications in Nonlinear Science and Numerical Simulation, 2017, 43, 369-384 65 D T Wood, H V Kojouharov, D T Dimitrov, Universal approaches to approximate biological systems with nonstandard finite difference methods, Mathematics and Computers in Simulation, 2017, 133, 337-350 66 R E Mickens, A B Gumel, Numerical study of a non-standard finite-difference scheme for the Van der Pol equation, Journal of Sound and Vabration, 2000, 250, 955-963 67 R E Mickens, Step-Size Dependence Of The Period For A Forward-Euler Scheme Of The Van Der Pol Equation, Journal of Sound and Vabration, 2002, 258, 199202 68 R E Mickens, A numerical integration technique for conservative oscillators combining nonstandard finite-difference methods with a Hamilton’ s principle, Journal of Sound and Vabration, 2005, 285, 477-482 69 D T Wood, Advancements and applications of nonstandard finite difference methods, The University of Texas at Arlington, https://rc.library.uta.edu/utair/handle/10106/25406 70 O F Egbelowo, The Nonstandard Finite Difference Method Applied to Pharmacokinetic Models, University of the Witwatersrand, Johannesburg, 2018, https://hdl.handle.net/10539/27306 71 G Gonzalez-Parra, A J Arenas, B M Chen-Charpentier, Combination of nonstandard schemes and Richardson’s extrapolation to improve the numerical solution of population models, Mathematical and Computer Modelling, 2010, 52, 1030-1036 168 72 J Martin-Vaquero, A Martin del Rey, A H Encinas, J D Hernandez Guillen, A Queiruga-Dios, G Rodriguez Sanchez, Higher-order nonstandard finite difference schemes for a MSEIR model for a malware propagation, Journal of Computational and Applied Mathematics, 2017, 317, 146-156 73 J Martin-Vaquero, A Queiruga-Dios, A Martin del Rey, A H Encinas, J D Hernandez Guillen, G Rodriguez Sanchez, Variable step length algorithms with high-order extrapolated non-standard finite difference schemes for a SEIR model, Journal of Computational and Applied Mathematics, 2018, 330, 848-854 74 R E Mickens, A nonstandard finite difference scheme for the diffusionless Burgers equation with logistic reaction, Mathematics and Computers in Simulation, 2003, 62, 117-124 75 R E Mickens, A nonstandard finite difference scheme for a Fisher PDE having nonlinear diffusion, Computers and Mathematics with Applications, 2003, 45, 429-436 76 R E Mickens, A nonstandard finite difference scheme for a PDE modeling combustion with nonlinear advection and diffusion, Mathematics and Computers in Simulation, 2005, 69, 439-446 77 R E Mickens, A nonstandard finite difference scheme for a nonlinear PDE having diffusive shock wave solutions, Mathematics and Computers in Simulation, 2001, 55, 549-555 78 M Chapwanya, J M.-S Lubuma, R E Mickens, Positivity-preserving nonstandard finite difference schemes for cross-diffusion equations in biosciences, Computers and Mathematics with Applications, 2014, 68, 1071-1082 79 A J Arenas, G González-Parra, B Melendez Caraballo, A nonstandard finite difference scheme for a nonlinear Black-Scholes equation, Mathematical and Computer Modeling, 2013, 57, 1663-1670 80 M Ehrhardt, R E.Mickens, A nonstandard finite difference scheme for convectiondiffusion equations having constant coefficients, Applied Mathematics and Computation, 2013, 219, 6591-6604 169 81 Y Yang, J Zhou, X Ma, T Zhang, Nonstandard finite difference scheme for a diffusive within-host virus dynamics model with both virus-to-cell and cell-tocell transmissions, Computers and Mathematics with Applications, 2016, 72, 1013-1020 82 A.J Arenas, G Gonzalez-Parra, B M Chen-Charpentier, Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order, Mathematics and Computers in Simulation, 2016, 121, 48-63 83 J Cresson, A Szafra´nska, Discrete and continuous fractional persistence problems-the positivity property and applications, Communications in Nonlinear Science and Numerical Simulation, 2017, 44, 424-448 84 K Moaddy, I Hashim, S Momani, Non-standard finite difference schemes for solving fractional order Rossler chaotic and hyperchaotic systems, Computers and Mathematics with Applications, 2011, 62, 1068-1074 85 A M Nagy, N H Sweilam, An efficient method for solving fractional HodgkinHuxley model, Physics Letters A, 2014, 378, 1980-1984 86 H Su, X Ding, Dynamics of a nonstandard finite-difference scheme for MackeyGlass system, Journal of Mathematical Analysis and Applications, 2008, 344, 932-941 87 Y Wang, Dynamics of a nonstandard finite-difference scheme for delay differential equations with unimodal feedback, Communications in Nonlinear Science and Numerical Simulation, 2012, 17, 3967-3978 88 S M Garba, A B Gumel, A S Hassan, J M S Lubuma, Switching from exact scheme to nonstandard finite difference scheme for linear delay differential equation, Applied Mathematics and Computation, 2015, 258, 388-403 89 S Elaydi, An introduction to Difference Equations, Springer-Verlag, 2005, New York 90 V Sundarapandian, Global asymptotic stability of nonlinear cascade systems, Applied Mathematics Letters, 2002, 15, 275-277 170 91 A Iggidr, M Bensoubaya, New Results on the Stability of Discrete-Time Systems and Applications to Control Problems, Journal of Mathematical Analysis and Applications, 1998, 219, 392-414 92 A Gerisch and R Weiner, The Positivity of Low-Order Explicit Runge-Kutta Schemes Applied in Splitting Methods, Computers and Mathematics with Applications, 2003, 45, 53-67 93 Z Horváth, Positivity of Runge-Kutta methods and diagonally split Runge-Kutta methods, Applied Numerical Mathematics, 1998, 28, 306-326 94 Z Horváth, On the positivity step size threshold of Runge-Kutta methods, Applied Numerical Mathematics, 2005, 53, 341-356 95 J F B M Kraaijevanger, Contractivity of Runge-Kutta methods, BIT, 1991, 31, 482-528 96 P M Manning, G F Margrave, Introduction to non-standard finite-difference modelling, CREWES Research Report, 2006, 16 97 J.E Keymer, P.A Marquet, J.X Velasco-Hernandez, S.A Levin, Extinction thresholds and metapopulation persistence in dynamic landscapes, The American Naturalist, 2000, 156, 478-494 98 P Amarasekare, H Possingham, Patch Dynamics and Metapopulation Theory: the Case of Successional Species, Journal of Theoretical Biology 2001, 209, 333-344 99 H.R Thieme, Convergence results and a Poincar Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 1992, 30, 755-763 100 J La Salle, S Lefschetz, Stability by Liapunov’s Direct Method, Academic Press, 1961, New York 101 J C Piqueira, V O Araujo, A modified epidemiological model for computer viruses, Applied Mathematics and Computation, 2009, 213, 355-360 171 102 P Szor, The art of computer virus research and defense, Addison-Wesley Education Publishers Inc, 2005 103 A A Berryman, The origins and evolution of predator-prey theory, Ecology, 1992, 73, 1530-1535 104 L Edelstein-Keshet, Mathematical Models in Biology, Society for Industrial and Applied Mathematics, 1998, Philadelphia 105 L M Ladino, E I Sabogal, J C Valverde, General functional response and recruitment in a predator-prey system with capture on both species, Mathematical Methods in the Applied Sciences, 2015, 38, 2876-2887 106 X Y Meng, H F Hou, H Xiang, Q Y Yin, Stability in a predator-prey model with Crowley-Martin function and stage structure for prey, Applied Mathematics and Computation, 2014, 232, 810-819 107 D T Dimitrov and H V Kojouharov, Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis functional response, Applied Mathematics and Computation, 2005, 162, 523-538 108 G J Cooper, J H Verner, Some Explicit Runge-Kutta Methods of High Order, SIAM Journal on Numerical Analysis, 1972, 9, 389-405 109 C M Kribs-Zaleta and J X Velasco-Hernández, A simple vaccination model with multiple endemic states, Mathematical Biosciences, 2000, 164, 183-201 110 R E Mickens, Exact solutions to a finite-difference model of a nonlinear reaction-advection equation: Implications for numerical analysis, Numerical Methods for Partial Differential Equations, 1989, 5, 313-325 111 R E Mickens, Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations, Journal of Difference Equations and Applications, 2005, 11, 645-653 112 P Seibert, R Suarez, Global stabilization of nonlinear cascade systems, Systems and Control Letters, 1990, 14, 347-352 172 ... 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