Phát Triển Phương Pháp Sai Phân Khác Thường Giải Một Số Lớp Phương Trình Vi Phân.pdf

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 SOM[.]

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 kxk The norm of the vector x 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 y(t), ˙ y ′ (t), dy(t)/dt The first derivative of the function y(t) vi x ≥ 0, y ≥ is a positively invariant set for the NSFD scheme (2.4.3) if α1 ≥ 0, α2 ≤ 0, α3 ≤ 0, α4 ≥ 0, α5 ≤ 0, β1 ≥ 0, β2 ≤ 0, β3 ≥ 0, β4 ≤ 0, β5 ≤ 0, α6 ≥ 0, (2.4.4) β6 ≥ Proof It is easy to transform the scheme (2.4.3) to the explicit form xk + ϕα1 xk r(xk ) − ϕα3 xk yk φ(xk ) − ϕα5 m1 xk , − ϕα2 r(xk ) + ϕα4 yk φ(xk ) + ϕα6 m1 yk + ϕβ1 yk s(yk ) + ϕβ3 cxk yk φ(xk ) − ϕβ5 m2 yk = G(xk , yk ) := − ϕβ2 s(yk ) − ϕβ4 cxk φ(xk ) + ϕβ6 m2 xk+1 = F (xk , yk ) := yk+1 (2.4.5) Since the parameters αj and βj (j = 1, , 6) satisfy (2.4.4), the positivity of the scheme is proved Proposition 2.3 The NSFD scheme (2.4.3) preserves the set of equilibrium points of system (2.4.1) Proof The theorem is proved easily by using the definition of equilibria of discrete dynamical systems (see [A5]) 2.4.3 Stability analysis Firstly, we study the LAS of the NSFD scheme (2.4.3) Through the rest of this subsection, we always assume that the condition (2.4.4) is satisfied The following theorem is proved by using Theorem 1.10 in combination with Theorem 1.13 Theorem 2.25 (LAS property of NSFD schemes) (i) The extinction equilibrium point P0∗ = (x∗0 , y0∗ ) = (0, 0) of the NSFD scheme (2.4.3) is LAS if m1 > r(0) and m2 > s(0), and unstable if m1 < r(0) or m2 < s(0) (ii) Consider the NSFD scheme (2.4.3) in the case m1 < r(0) and m2 > s(0) + cKφ(K) and assume that T1 := 2α6 m1 − 2α2 r(K) + Kr′ (K) > 0, (2.4.6) T2 := s(0) − m2 + cKφ(K) − 2β2 s(0) − 2β4 cKφ(K) + 2β6 m2 > Then, the equilibrium point of the form P1∗ = (K, 0) is LAS Moreover, P1∗ is and unstable if m1 ≥ r(0) or m2 < s(0) + cKφ(K) 91 (iii) Consider the NSFD scheme (2.4.3) in the case m1 > r(0) − M φ(0) and m2 < s(0) and assume that T3 := r(0) − M φ(0) − m1 − 2α2 r(0) + 2α4 M φ(0) + 2α6 m1 > 0, T4 := M s′ (M ) − 2β2 s(M ) + 2β6 m2 > (2.4.7) Then, the equilibrium point of the form P2∗ = (0, M ) is LAS Moreover, P2∗ is unstable if m1 < r(0) − M φ(0) or m2 ≥ s(0) (iv) Suppose that the equilibrium point of the form P3∗ = (x∗ , y ∗ ) belongs to Ω Consider the NSFD scheme (2.4.3) under the assumption T5 := − x∗ [r′ (x∗ ) − y ∗ φ′ (x∗ )][−β2 s(y ∗ ) − β4 cx∗ φ(x∗ ) + β6 m2 ] − y ∗ s′ (y ∗ )[−α2 r(x∗ ) + α4 y ∗ φ(x∗ ) + α6 m1 ] − x∗ y ∗ s′ (y ∗ )[r′ (x∗ ) − y ∗ φ′ (x∗ )] − cx∗ y ∗ φ(x∗ )[φ(x∗ ) + x∗ φ′ (x∗ )] > 0, T6 := − α2 r(x∗ ) + α4 y ∗ φ(x∗ ) + α6 m1 + x∗ [r′ (x∗ ) − y ∗ φ′ (x∗ )] > 0, T7 := − β2 s(y ∗ ) − β4 cx∗ φ(x∗ ) + β6 m2 + y ∗ s′ (y ∗ ) > (2.4.8) Then, P3∗ is LAS It is important to note that Theorem 2.25 only confirms the LAS of NSFD schemes, meanwhile, P0∗ = (0, 0) is not only LAS but also GAS To investigate the GAS of this equilibrium point, we use the Lyapunov direct method with the help of the following family of Lyapunov functions V (xk , yk ) := αxk yk + βx2k + γxk + δyk , (xk , yk ) ∈ R2+ , (2.4.9) where α, β, γ, δ > are positive parameters that will be selected so that the function V (xk , yk ) satisfies Theorem 1.9 Obviously, the function V (xk , yk ) defined by (2.4.9) is continuous on R2+ , and V (xk , yk ) → ∞ as ||(xy , yk )|| → ∞ Moreover, V (P0∗ ) = 0, and V (xk , yk ) > for any (xk , yk ) ∈ R2+ , (xk , yk ) 6= (0, 0) Therefore, we only need to determine conditions guaranting that ∆V (xk , yk ) = V (xk+1 , yk+1 ) − V (xk , yk ) < 0, 92 ∀(xk , yk ) ∈ R2+ /{(0, 0)} After a lot of algebraic transformations (see [A5]), we conclude that the function V (xk , yk ) defined by (2.4.9) satisfies ∆V (xk , yk ) < for all (xk , yk ) ∈ R2+ /{(0, 0)} if  −α2 cr(0) + α6 cm1 β max c, < , β6 m2 α cα4 φ(0) α < , α4 + β4 < β6 m2 δ   −α2 cr(0) + cα6 m1 max c, β6 m2  < γ , δ (2.4.10) If the scheme (2.4.3) is fixed, then the selection of the parameters α, β, γ, δ > satisfying the above relations is completely feasible Therefore, we have the following result Theorem 2.26 Consider the NSFD scheme (2.4.3) in the case m1 ≥ r(0) and m2 ≥ s(0) and assume that α4 + β4 < (2.4.11) Then, the extinction equilibrium point P0∗ = (0, 0) is GAS 2.4.4 Dynamically consistent NSFD schemes Theorem 2.27 The NSFD scheme (2.4.3) is dynamically consistent with the model (2.4.1) if the parameters αj , βj (j = 1, , 6) satisfy the conditions listed in Table 2.2, where the columns list sufficient conditions for the scheme (2.4.3) to preserve corresponding properties of the model (2.4.1) for different cases of the parameters The symbol ′′ ∗′′ means that the set of equilibrium points of the model (2.4.1) is always preserved by the scheme (2.4.3) 93 Table 2.2 The sufficient conditions for dynamic consistency Set of equilibria Positivity Stability m1 ≥ r(0) and m2 ≥ s(0) * (2.4.4) (2.4.11) m1 < r(0) and m2 > s(0) + cKφ(K) * (2.4.4) (2.4.6) m1 > r(0) − M φ(0) and m2 < s(0) * (2.4.4) (2.4.7) m1 < r(0) − M φ(0) and m2 < s(0) * (2.4.4) (2.4.8) m1 < r(0) and s(0) < m2 < s(0) + cKφ(K) * (2.4.4) (2.4.8) (m1 , m2 ) Remark 2.8 There are infinitely many ways for selecting the parameters αj , βj (j = 1, , 6) satisfying the conditions listed in Table 2.2 This shows the existence of dynamically consistent NSFD schemes for the model (2.4.1) 2.4.5 Numerical simulation In this section, we perform some numerical simulations to confirm the validity of the theoretical results Also, the numerical simulations demonstrate the advantages of NSFD schemes over SFD ones Example 2.11 (Dynamics of SFD schemes and NSFD schemes) Let us reconsider the example presented in [105] In this example, we have xr(x) = 15x , x + 10 ys(y) = 5y , y + 10 xφ(x) = x , x + 30 c = 0.003, and for cases of the parameters (m1 , m2 ) in Corollary in [105], namely, (i) m1 = 1.53, m2 = 0.622 (iii) m1 = 1.4925, (v) m1 = 0.3, (ii) m1 = 1.53, m2 = 0.4789 m2 = 0.501 (iv) m1 = 1.38, (vi) m1 = 1.38, 94 m2 = 0.4789 m2 = 0.4789 m2 = 0.622 Firstly, we use the standard Euler, RK2 and Rk4 schemes for numerical simulation Numerical solutions obtained by these schemes in Case (i) of the parameters m1 and m2 are depicted in Figures 2.25-2.28 From these figures, we see that the positivity and stability of the continuous model are destroyed The numerical experiments in other cases of the parameters are analogous 600 500 400 300 x 200 100 -100 -200 -300 -400 20 40 60 80 100 120 140 160 180 200 t Figure 2.25 The x-component obtained by the explicit Euler scheme for (x0 , y0 ) = (100, 160), h = 1.111 after 180 iterations 95 160 (x, y) P* = (0, 0) 140 120 y 100 80 60 40 20 -400 -300 -200 -100 100 200 300 400 500 600 x Figure 2.26 The phase portrait obtained by the explicit Euler scheme for (x0 , y0 ) = (100, 160), h = 1.111 after 180 iterations 100 80 60 40 x 20 -20 -40 -60 -80 -100 20 40 60 80 100 120 140 160 180 200 t Figure 2.27 The x-component obtained by the RK4 scheme for (x0 , y0 ) = (100, 160), h = 1.429 after 140 iterations 96 160 140 (x, y) P* = (0, 0) 120 y 100 80 60 40 20 -100 -80 -60 -40 -20 20 40 60 80 100 x Figure 2.28 The phase potrait obtained by the RK4 scheme for (x0 , y0 ) = (100, 160), h = 1.429 after 140 iterations For cases of the parameters (m1 , m2 ) listed above, we consider NSFD schemes defined by Theorem 2.27 as follows Scheme (2.4.3)-(i): α1 = 1.2, α3 = −0.2, α5 = −0.2, αj+1 = − αj , j = 1, 3, 5, β1 = 1.2, β3 = 2.5, β5 = −0.1, βj+1 = − βj , j = 1, 3, 5, ϕ(h) = − e−h Scheme (2.4.3)-(ii): α1 = 1.16, α3 = −0.16, α5 = −0.1, αj+1 = − αj , j = 1, 3, 5, β1 = 1.2, β3 = 1.2, β5 = −0.1, βj+1 = − βj , j = 1, 3, 5, ϕ(h) = − e−h Scheme (2.4.3)-(iii): α1 = 1.16, α3 = −0.15, α5 = −0.2, αj+1 = − αj , j = 1, 3, 5, β1 = 1.1, β3 = 1.16, β5 = −0.16, βj+1 = − βj , j = 1, 3, 5, ϕ(h) = − e−h Scheme (2.4.3)-(iv): α1 = 1.2, α3 = −0.25, α5 = −0.1, αj+1 = − αj , j = 1, 3, 5, β1 = 1.2, β3 = 1.1, β5 = −0.2, βj+1 = − βj , j = 1, 3, 5, ϕ(h) = − e−h Scheme (2.4.3)-(v): α1 = 1.25, α3 = −0.1, α5 = −0.2, αj+1 = − αj , j = 1, 3, 5, β1 = 1.16, β3 = 1.25, β5 = −0.15, βj+1 = − βj , j = 1, 3, 5, ϕ(h) = − e−h 97 Scheme (2.4.3)-(vi): α1 = 1.25, α3 = −0.2, α5 = −0.2, αj+1 = − αj , j = 1, 3, 5, β1 = 1.25, β3 = 1.36, β5 = −0.16, βj+1 = − βj , j = 1, 3, 5, ϕ(h) = − e−h Numerical solutions obtained by these NSFD schemes are depicted in Figures 2.292.34 In these figures, each blue curve represents a trajectory corresponding to a specific initial data From these figures, we see that all of the essential properties of the continuous model are preserved Additionally, the results agree with the numerical simulations presented in [105, Section 4] 160 140 120 y 100 80 60 40 20 P0 0 10 20 30 40 50 60 70 80 90 100 x Figure 2.29 The phase portrait obtained by the scheme (2.4.3)-(i) for h = 2.5, t ∈ [0, 2000] 98 0.9 0.8 0.7 y 0.6 0.5 P* 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x Figure 2.30 The phase portrait obtained by the scheme (2.4.3)-(ii) for h = 2.5, t ∈ [0, 2000] and P2∗ = (0, 0.4406) 10 y P* 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 x Figure 2.31 The phase portrait obtained by the scheme (2.4.3)-(iii) for h = 2.5, t ∈ [0, 2000] and P2∗ = (0, 0.4406) 99 0.9 0.8 0.7 y 0.6 0.5 P* 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x Figure 2.32 The phase portrait obtained by the scheme (2.4.3)-(iv) for h = 2.5, t ∈ [0, 2000] and P3∗ = (0.7575, 0.4422) 10 y 10 P* 20 30 40 50 60 70 80 90 100 x Figure 2.33 The phase portrait obtained by the scheme (2.4.3)-(v) for h = 2.5, t ∈ [0, 2000] and P3∗ = (39.996, 0.0143) 100 10 y 0 P* 10 20 30 40 50 60 70 80 90 100 x Figure 2.34 The phase portrait obtained by the scheme (2.4.3)-(vi) for h = 2.5, t ∈ [0, 2000] and P1∗ = (0.8696, 0) Example 2.12 (Computational time and errors of the SFD and NSFD schemes) In this example, we report some numerical examples to show advantages of NSFD schemes over the SFD ones from the point of view of computational time For this purpose, we reconsider the model (2.4.1) with the parameters given in Case (i) of Example 2.11, that is: m1 = 1.53, m2 = 0.622, t ∈ [0, 200] We will use the standard Euler, RK2 and RK4 schemes and the NSFD scheme 3-(i) with ϕ(h) = h to solve the model Because it is impossible to find the exact solution of the model, we take the numerical solution obtained from the RK4 scheme with a very small step size, namely, h = 10−4 , as the benchmark solution Moreover, we use the formula n o error := max |xk − x(tk )| + |yk − y(tk )| , k
as a measure of accuracy of the schemes, where (xk , yk ) and x(tk ), y(tk ) are com- puted solutions by the numerical schemes and the benchmark solution, respectively The comparison of the errors of the schemes are given in Table 2.3 101 Table 2.3 The errors of the numerical schemes Step size Euler scheme RK2 scheme RK4 scheme NSFD scheme 0.8 133.8659 108.1124 189.8346 45.8741 1.7255e + 04 2.7334e + 03 778.1996 50.1038 1.6 2.7720e + 22 8.5512e + 25 109.5938 57.5249 5.6193e + 33 1.4965e + 44 8.1025e + 19 60.8888 From this table, we see that the errors of SFD schemes increase very fast w.r.t the increase of the step sizes and become enormous for h ≥ 0.8 Therefore, the solutions obtained by SFD schemes are meaningless Meanwhile, the NSFD scheme yields smaller errors Figure 2.35 shows the numerical solutions obtained by the schemes for h = 0.8 From this figure, it is clear that NSFD scheme behaves as the benchmark solution, meanwhile, the SFD schemes destroy the positivity and their solutions strongly different from the benchmark solution Thus, the advantages of the NSFD schemes over the SFD schemes are shown 160 140 Euler scheme RK2 scheme RK4 scheme NSFD scheme Benchmark solution 120 y 100 80 60 40 20 -200 -150 -100 -50 50 100 150 x Figure 2.35 The numerical solutions obtained by the numerical schemes in Example 2.12 102 We now consider the computational time of the schemes by using MATLAB R2014b on 32-bit Operating System with Intel(R) Core(TM) i5-2430M, 2.40GHz, GB RAM The computational time of the schemes is given in Table 2.4, and the graphs of these errors are depicted in Figure 2.36 From these results, we see that the Euler scheme is a bit faster than the NSFD scheme The reason is that the Euler scheme is constructed by the local approximation of the right-hand side function, meanwhile, the NSFD scheme is constructed by the non-local approximation, and therefore, the iterative formula of the NSFD scheme is more complicated However, it should be emphasized that the Euler scheme provides the numerical solutions which cannot preserve the properties of the continuous model Also, from Table 2.4 it is clear that the RK2 and RK4 schemes are slower than the NFSD scheme This may be explained that these schemes require much time to evaluate the right-hand side function It is more important that both schemes provide the numerical solutions of bad quality as it is seen in Figure 2.35 It is worth to recall that one of the advantage of the NSFD schemes is the exact preservation of the essential properties of the continuous model for any finite step size Therefore, they are very effective when studying the continuous model in a long time Meanwhile, for large step sizes, SFD schemes can provide the solutions which are completely different from the solutions of the continuous model Table 2.4 The time of the schemes in seconds Step size Euler scheme RK2 scheme RK4 scheme NSFD scheme 7.4034e − 04 0.0198 0.0354 7.7383e − 04 1.6 8.5133e − 04 0.0229 0.0446 8.8545e − 04 0.0013 0.0362 0.0704 0.0014 0.8 0.0016 0.0454 0.0882 0.0017 103 1.7 ×10 -3 0.09 Euler scheme NSFD scheme RK2 scheme RK4 scheme 1.6 0.08 1.5 0.07 1.4 0.06 Time Time 1.3 1.2 0.05 1.1 0.04 0.03 0.9 0.02 0.8 0.7 0.8 1.2 1.4 1.6 1.8 0.01 0.8 1.2 h 1.4 1.6 1.8 h Figure 2.36 Computational time of the numerical schemes in seconds with h = 0.8 in Example 2.12 Example 2.13 (Comparison with numerical solutions by the ode45 code in MATLAB) In this example, we will compare the numerical solutions obtained by the ode45 code and NSFD schemes For this purpose, we consider the model with the parameters m1 = 20, m2 = 0.5, c = 0.003, t ∈ [0, 100], and the initial condition x(0) = 10−6 , y(0) = It is well known that the ode45 code is very popular and is based on the Dormand-Prince (4, 5) pair [3, 19, 20] The ode45 is operated completely differently from the Euler, RK2, RK4 and NSFD schemes, namely, it uses variable step sizes instead of uniform step sizes Therefore, the error can be controlled at each iteration step However, as will be seen below, although the error can be controlled, the positivity, stability and monotonicity of the continuous model are still destroyed by the ode45 Indeed, from Figure 2.37 which plots the numerical solution obtained by the ode45, it is clear that the positivity, stability and monotonicity of the solution are destroyed In 104 this case, the ode45 requires 2236 grid points with hmin = 0.0230 and hmax = 0.0550 The graph of the required step sizes versus iterations is depicted in Figure 2.38 The computational time is 0.0753 seconds Together with the ode45 solution, the numerical solution obtained by NSFD scheme with ϕ(h) = h and h = is also depicted in Figure 2.37 From this figure, it is observed that the positivity, stability and monotonicity of the solution are reflected exactly Especially, the computational time of the NSFD scheme is only 1.0330e − 04 seconds, which is much less than that of the ode45 Thus, from this example, we conclude some advantages of the NSFD schemes compared with the ode45 1.5 ×10 -6 ODE45 NSFD scheme x 0.5 -0.5 -1 10 20 30 40 50 60 70 80 90 100 t Figure 2.37 Numerical solutions obtained the ode45 and NSFD scheme The ode45 requires 2236 grid points with hmin = 0.0230 and hmax = 0.0550 and the computational time is 0.0753 seconds NSFD scheme 3-(i) use ϕ(h) = h and h = 1, the computational time is 1.0330e − 04 seconds 105