58 Solving Oscillating Problems Using Modifying Runge-Kutta Methods Zainab Khaled Ghazal Kasim Abbas Hussain Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq. kasimabbas@uomustansiriyah.edu.iq zkgzainab@gmail.com Abstract This paper develop conventional Runge-Kutta methods of order four and order five to solve ordinary differential equations with oscillating solutions. The new modified Runge- Kutta methods (MRK) contain the invalidation of phase lag, phase lag’s derivatives, and amplification error. Numerical tests from their outcomes show the robustness and competence of the new methods compared to the well-known Runge-Kutta methods in the scientific literature. Keywords: Explicit Runge-Kutta methods, oscillating problems Phase-lag properties 1. Introduction This essay dealing with the system of special first-order ordinary differential equation in the following form: 𝑒′(𝑑) = 𝑓(𝑑, 𝑒), 𝑒(𝑑0) = 𝑒0. (1) Such problems are often observed in various applied sciences, such as quantum chemistry, astronomy, quantum mechanics, electronics, elastics, and chemical physics (see [1,2]). Conventionally equation (1) is solved by using Runge-Kutta (RK) methods or two-step methods [3]. Several authors presented optimized numerical methods based on the phase lag characteristics (such as [4,5,6]). Exact and numerical solutions produce an angle between them is known as Phase-lag, and the distance of the numerical solution from the periodic solution is called amplification error. A five-stage four order Runge-Kutta method with phase-fitted and amplification-fitted proposed by Adel et al. [7]. Hussain et al. [8] proposed an optimized Runge-Kutta method to solve problem (1). The phase lag has the feature to develop more accurate numerical methods. This motivates us to propose the Runge-Kutta Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/34.4.2703 Article history: Received 11 April 2021, Accepted 30 May 2021, Published in October 2021. mailto:kasimabbas@uomustansiriyah.edu.iq %20zkgzainab@gmail.com Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 59 methods by using phase lag properties. Based on the previous work, we present a modified approach for the Runge–Kutta methods by combining the annulment of phase lag, phase lag's derivative, and amplification factor. The new MRK methods have been used to solve oscillation problems. The paper is orderly as follows: In Section 2, the phase-lag properties of the MRK method are given. In Section 3, we present how the new methods are derived. In section 4, we offer numerical results to illustrate the efficiency of the new MRK method. Section 5 is dedicated to Conclusions. 2. Phase Lag Properties of MRK Methods We are concerned with the numerical solution of the ODEs of the form (1) with an oscillating solution. The general form of explicit MRK method for solving (1) is defined as follows: 𝑒𝑛+1 = 𝑒𝑛 + βˆ‘ 𝑏𝑖 𝑓(𝑑𝑛 + 𝑐𝑖 , π‘ˆπ‘– ) 𝑠 𝑖=1 (2) π‘ˆπ‘– = 𝑑𝑖 𝑒𝑛 + β„Ž βˆ‘ π‘Žπ‘–π‘— π‘–βˆ’1 𝑗=1 𝑓(𝑑𝑛 + 𝑐𝑖 β„Ž, π‘ˆπ‘— ), 𝑖 = 1,2, . . . , 𝑠. (3) The corresponding Butcher tableau for equations (2) – (3) is expressed as follows Table 1. S-stage modified MRK method 0 𝑐2 𝑑2 π‘Ž21 𝑐3 𝑑3 π‘Ž31 π‘Ž32 . . . . . . . . . . . . . . . 𝑐𝑠 𝑑𝑠 π‘Žπ‘ 1 π‘Žπ‘ 2 … π‘Žπ‘ π‘ βˆ’1 𝑏1 𝑏2 … π‘π‘ π‘ βˆ’1 𝑏𝑠 Constructing the new method depends on phase-lag and dissipation analysis proposed in [9]. To this goal, we use the following test equation, 𝑒′ = 𝑖𝑀𝑒, 𝑀 ∈ 𝑅 (4) An implementation of MRK method (2)-(3) to the test equation (4) we yield 𝑒𝑛 = π‘žβˆ— 𝑛 𝑒0, π‘žβˆ— 𝑛 = 𝑆(𝑧2) + 𝑖𝑧 𝑉(𝑧2) (5) where 𝑧 = π‘€β„Ž and 𝑆, 𝑉 are polynomials in 𝑧2 entirely specified by the coefficients π‘Žπ‘–π‘— , 𝑐𝑖 and 𝑏𝑖 of MRK method (2)-(3). The comparison between equations (5) and (4) produces the following definition. Definition: [9] The following quantities in RK method (2)-(3) 1- 𝑃(𝑧) = 𝑧 βˆ’ arg[π‘žβˆ—(𝑧)] = 𝑧 βˆ’ tan βˆ’1 ( 𝑧.𝑉(𝑧 2) 𝑆(𝑧 2) ) , 2- 𝐷(𝑧) = 1 βˆ’ |π‘žβˆ—(𝑧)| = 1 βˆ’ √(𝑆 2(𝑧2) + 𝑧2 𝑉2(𝑧2)). are respectively called phase lag or dispersion error and the amplification factor or dissipation error. If 𝑃(𝑧) = Ο(π‘§π‘Ÿ+1) and 𝐷(𝑧) = Ο(𝑧𝑝+1), then the method is said to be of dispersion order π‘Ÿ and dissipation order 𝑝 . 3. Derivation of new MRK methods This section proceeds to derive the modified RK method by abolishing phase lag, phase lag’s derivative, and amplification error. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 60 3.1 Five-Stage Fourth-Order Method A fourth-order MRK method of five-stage is given as follows [10]: Table 2. Five-stage modified explicit Runge-Kutta method To construct a modified RK method, we set 𝑑2, 𝑑3, and 𝑑4 as free parameters and the other parameters are the same as in Table 2. Motivated by the approach in [8], we obtain the phase lag, phase lag derivative and the amplification factor, which depend on 𝑑2, 𝑑3 and 𝑑4 as follow: 𝑃(𝑧) = tan(𝑧) ( 1 24 𝑑2 𝑧 4 + (βˆ’ 5 16 𝑑3 βˆ’ 5 96 𝑑4 βˆ’ 13 96 ) 𝑧2 + 1) βˆ’ 𝑧 ( 1 120 𝑧4 + (βˆ’ 1 16 βˆ’ 5 48 𝑑3) 𝑧 2 + 7 32 + 25 48 𝑑3 + 25 96 𝑑4), (6) 𝑃′(𝑧) = (1 + π‘‘π‘Žπ‘›2(𝑧)) ( 1 24 𝑑2𝑧 4 + (βˆ’ 5 16 𝑑3 βˆ’ 5 96 𝑑4 βˆ’ 13 96 ) 𝑧2 + 1) + tan(𝑧) ( 1 6 𝑑2𝑧 3 + 2 (βˆ’ 5 16 𝑑3 βˆ’ 5 96 𝑑4 βˆ’ 13 96 ) 𝑧) βˆ’ 1 120 𝑧4 βˆ’ (βˆ’ 1 16 βˆ’ 5 48 𝑑3) 𝑧 2 βˆ’ 7 32 βˆ’ 25 48 𝑑3 βˆ’ 25 96 𝑑4 βˆ’ 𝑧 ( 1 30 𝑧3 + 2 (βˆ’ 1 16 βˆ’ 5 48 𝑑3) 𝑧), (7) 𝐷(𝑧) = 1 14400 𝑧10 + (βˆ’ 1 960 + 1 576 𝑑2 2 βˆ’ 1 576 𝑑3) 𝑧 8 + ( 5 1152 𝑑4 + 29 3840 + 25 1152 𝑑3 + 25 2304 𝑑3 2 βˆ’ 5 1152 𝑑2𝑑4 βˆ’ 13 1152 𝑑2 βˆ’ 5 192 𝑑2𝑑3) 𝑧 6 + (βˆ’ 25 1152 𝑑3𝑑4 + 25 9216 𝑑4 2 + 1 12 𝑑2 βˆ’ 25 2304 𝑑3 2 βˆ’ 83 9216 βˆ’ 85 4608 𝑑4 βˆ’ 5 192 𝑑3) 𝑧 4 + (βˆ’ 685 3072 βˆ’ 305 768 𝑑3 + 625 9216 𝑑4 2 + 625 2304 𝑑3𝑑4 + 625 2304 𝑑3 2 + 5 512 𝑑4) 𝑧 2. (8) Now, solving equations (6),(7), and (8), we get the values of free parameters in terms of 𝑧, and the expressions are too complicated, we use the following Taylor series expressions, 𝑑2 = 1 βˆ’ 29 630 𝑧2 βˆ’ 311 75600 𝑧4 βˆ’ 46369 74844000 𝑧6 βˆ’ 120976607 1225944720000 𝑧8 βˆ’ 0 1 5 𝑑2 1 5 2 5 𝑑3 0 2 5 4 5 𝑑4 6 5 βˆ’ 12 5 2 1 1 βˆ’ 17 8 5 βˆ’ 5 2 5 8 13 96 0 25 48 25 96 1 12 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 61 1756548533 110335024800000 𝑧10 βˆ’ 3475880890247 1350500703552000000 𝑧12 βˆ’ β‹―, 𝑑3 = 1 βˆ’ 4 1575 𝑧4 βˆ’ 47 94500 𝑧6 βˆ’ 1009 13365000 𝑧8 1695769 139311900000 𝑧10 βˆ’ 271269671 137918781000000 𝑧12 βˆ’ β‹―, 𝑑4 = 1 βˆ’ 8 1575 𝑧4 βˆ’ 37 47250 𝑧6 βˆ’ 437 11694375 𝑧8 4558639 766215450000 𝑧10 βˆ’ 32225033 34479695250000 𝑧12 βˆ’ β‹―. (9) It can observe that for 𝑧 β†’ 0, the new method reduces to an original RK method given in Dormand [10]. 3.2 Six-Stage Five-Order Method Consider the following six-stage modified RK method which can be expressed in Butcher tableau [3]. Table 3. Five-stage modified explicit Runge-Kutta method According to the method in Table 3, the phase lag, phase lag derivative, and amplification factor become 𝑃(𝑧) = tan(𝑧) (1 + (βˆ’ 31 90 βˆ’ 1 15 𝑑4 βˆ’ 4 45 𝑑5) 𝑧 2 + 1 24 𝑧4 βˆ’ 1 1280 𝑧6) βˆ’ 𝑧 ( 7 90 𝑑6 + + 13 30 + 2 15 𝑑4 + 16 45 𝑑5 + (βˆ’ 7 90 βˆ’ 1 20 𝑑4) 𝑧 2 + 1 120 𝑧4) (10) 𝑃′(𝑧) = (1 + π‘‘π‘Žπ‘›2(𝑧)) (1 + (βˆ’ 31 90 βˆ’ 1 15 𝑑4 βˆ’ 4 45 𝑑5) 𝑧 2 + 1 24 𝑧4 βˆ’ 1 1280 𝑧6) + tan(𝑧) (2 (βˆ’ 31 90 βˆ’ 1 15 𝑑4 βˆ’ 4 45 𝑑5) 𝑧 + 1 6 𝑧3 βˆ’ 3 640 𝑧5) βˆ’ 7 90 𝑑6 βˆ’ 13 30 βˆ’ 2 15 𝑑4 βˆ’ 16 45 𝑑5 βˆ’ (βˆ’ 7 60 βˆ’ 1 20 𝑑4) 𝑧 2 βˆ’ 1 120 𝑧4 βˆ’ 𝑧 ( 1 30 𝑧3 + 2 (βˆ’ 7 60 βˆ’ 1 20 𝑑4) 𝑧), (11) 0 1 4 1 1 4 1 4 1 1 8 1 8 1 2 𝑑4 0 0 1 2 3 4 𝑑5 3 16 βˆ’ 3 8 3 8 9 16 1 𝑑6 βˆ’ 3 7 8 7 6 7 βˆ’ 12 7 8 7 7 90 0 16 45 2 15 16 45 7 90 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 62 𝐷(𝑧) = 1 1638400 𝑧12 + 1 230400 𝑧10 + ( 19 57600 βˆ’ 7 9600 𝑑4 + 1 7200 𝑑5) 𝑧 8 + ( 1 120 𝑑4 βˆ’ 1 675 𝑑5 + 1 400 𝑑4 2 βˆ’ 163 17280 + 7 5400 𝑑6) 𝑧 6 + (βˆ’ 77 2700 𝑑4 βˆ’ 2 225 𝑑4 2 βˆ’ 44 2025 𝑑5 + 817 8100 βˆ’ 49 2700 𝑑6 βˆ’ 7 900 𝑑4𝑑6 βˆ’ 16 675 𝑑4𝑑5 + 16 2025 𝑑5 2) 𝑧4 + ( 88 675 𝑑5 + 4 225 𝑑4 2 + 49 8100 𝑑6 2 + 256 2025 𝑑5 2 + 64 675 𝑑4𝑑5 + 112 2025 𝑑5𝑑6 + 14 675 𝑑4𝑑6 + + 91 1350 𝑑6 βˆ’ 4 225 𝑑4 βˆ’ 451 900 ) 𝑧2 (12) Solving equations (10),(11), and (12), we find 𝑑4 = 1 βˆ’ 25 2016 𝑧4 βˆ’ 31 4536 𝑧6 βˆ’ 613 221760 𝑧8 βˆ’ 116287 103783680 𝑧10 βˆ’ 5938297 1307643680 𝑧12 βˆ’ 3273113521 17784371404800 𝑧14 βˆ’ β‹―, 𝑑5 = 1 + 347 21504 𝑧4 + 67 13824 𝑧6 + 7367 3548160 𝑧8 + 2819 3354624 𝑧10 + 1827169 5364817920 𝑧12 + 13092454033 94849980825600 𝑧14 + β‹―, 𝑑6 = 1 βˆ’ 247 4704 𝑧4 βˆ’ 1775 84672 𝑧6 βˆ’ 14143 1552320 𝑧8 βˆ’ 447763 121080960 𝑧10 βˆ’ 22865693 15256200960 𝑧12 βˆ’ 2520658027 4149686661120 𝑧14 βˆ’ β‹―. (13) It can demonstrate that for 𝑧 β†’ 0, the new method same as the original RK method given in Butcher [3]. 4. Numerical Results To evaluate the performance of the new modified Runge-Kutta methods suggested in this paper, we apply them to five oscillatory problems and then compared the numerical results with the several well-known efficient methods. We use the criteria of absolute error to measure the accuracy of the method, which is given by Absolute error = max (|𝑦(𝑑𝑛) βˆ’ 𝑦𝑛 |). Where 𝑦(𝑑𝑛) is the true solution and 𝑦𝑛 is the numerical solution. Figures 1-5 demonstrate the efficiency graphs of Log 10 (Max Error) versus step size β„Ž. Integration interval is [0, 1000] for all problems with step sizes β„Ž = 0.1 2𝑖 , 𝑖 = 1,2,3,4 ⁄ . The following numerical methods are used in the comparison. (i) ο‚· MRK4: modified five-stage fourth-order RK method presented in Table 1 and Eqs. (9) in Section 3 in this paper. ο‚· RK4D: the classical five-stage RK method of order four given in [10]. ο‚· RK4PF: the phase-fitted and amplification-fitted RK method given in [7]. ο‚· RK5S: six-stage RK method of order five given in [11]. ο‚· RK4MS: the modified four-order RK method given in [9]. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 63 ο‚· RK4B: the five-stage RK method of fourth-order given in [3]. (ii) ο‚· MRK5: modified six-stage fifth-order RK method presented in Table 2 and Eqs. (13) in Section 3 in this paper. ο‚· RK5B: the classical six-stage RK method of order five given in [3]. ο‚· ORK5T: Optimized fifth-order RK method given in [4]. ο‚· RK5M: Optimized RK method of order five given in [12]. ο‚· ORK5A: Optimized RK method of order five given in [5]. ο‚· RK5K: Optimized fifth-order RK method proposed in [6]. Problem 1: [13] 𝑒′′(𝑑) = 100 𝑒(𝑑) + 99 sin(𝑑) , 𝑒(0) = 1, 𝑒′(0) = 11. Whose exact solution is 𝑒(𝑑) = cos(10𝑑) + sin(10𝑑) + sin(𝑑) , and 𝑀 = 10. Problem 2: [14] 𝑒′′(𝑑) = βˆ’64 𝑒(𝑑), 𝑒(0) = 1, 𝑒′(0) = βˆ’2. Exact solution is 𝑒(𝑑) = 1 4 sin(8𝑑) + cos(8𝑑) , and 𝑀 = 8. Problem 3: [15] 𝑒1 β€²β€²(𝑑) + 𝑒1(𝑑) = 0.001 cos(𝑑) , 𝑒1(0) = 1, 𝑒1 β€² (0) = 0, 𝑒2 β€²β€²(𝑑) + 𝑒2(𝑑) = 0.001 sin(𝑑) , 𝑒2(0) = 0, 𝑒2 β€² (0) = 0.9995. Exact solution is 𝑒1(𝑑) = cos(𝑑) +0.0005 𝑑 sin(𝑑), 𝑒2(𝑑) = sin(𝑑) βˆ’0.0005 𝑑 cos(𝑑), and 𝑀 = 1. Problem 4: [16] 𝑒′′(𝑑) + ( 101 2 βˆ’ 99 2 βˆ’ 99 2 101 2 ) 𝑒(𝑑) = ( 93 2 cos(2𝑑) βˆ’ 99 2 sin(2𝑑) 93 2 sin(2𝑑) βˆ’ 99 2 cos(2𝑑) ), 𝑒(0) = ( 0 1 ) , 𝑒′(0) = ( βˆ’10 12 ). Exact solution and frequency are 𝑒(𝑑) = ( βˆ’ cos(10𝑑)(𝑑) βˆ’ sin(10𝑑) + cos(2𝑑) cos(10𝑑) + sin(10𝑑) + sin(2𝑑) ) , 𝑀 = 10. Problem 5: [17] Oscillatory system problem 𝑒′′(𝑑) + ( 13 βˆ’12 βˆ’12 13 ) 𝑒(𝑑) = ( 9 cos(2𝑑) βˆ’ 12 sin(2𝑑) βˆ’12 cos(2𝑑) + 9 sin(2𝑑) ), Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 64 𝑒(0) = ( 1 0 ) , 𝑒′(0) = ( βˆ’4 8 ). Exact solution and frequency are 𝑒(𝑑) = ( sin(𝑑) βˆ’ sin(5𝑑) + cos(2𝑑) sin(𝑑) + sin(5𝑑) + sin(2𝑑) ) , 𝑀 = 5. (a) Comparisons of the methods (i) in Section 4. (b) Comparisons of the methods (ii) in Section 4. Figure 1. The competence curves for Problem 1. (a) Comparisons of the methods (i) in Section 4 (b) Comparisons of the methods (ii) in Section 4. Figure 2. The competence curves for Problem 2. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 65 (a) Comparisons of the methods (i) in Section 4. (b) Comparisons of the methods (ii) in Section 4. Figure 3. The competence curves for Problem 3. (a) Comparisons of the methods (i) in Section 4. (b) Comparisons of the methods (ii) in Section 4. Figure 4. The competence curves for Problem 4. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 66 (a) Comparisons of the methods (i) in Section 4. (b) Comparisons of the methods (ii) in Section 4. Figure 5: The competence curves for Problem 5. 5. Conclusion Classical Runge-Kutta methods are adjusted for solving the system of first-order ordinary differential equations whose solutions have a marked periodic style in this paper. The new methods are based on a fourth and fifth algebraic order given in [10] and [3], respectively. The proposed methods have nullified the phase-lag, amplification error, and phase lag’s derivative. The results demonstrate the competence and effectiveness of the newly constructed methods, while they are compared to the standard methods and efficient RK methods. It is clear that the new MRK methods are the most accurate, and especially other than existing RK methods in the scientific literature. References 1. Mahmood, B. S.; Gaftan, A. M.; Fawzi, F. A. Alternating Directions Implicit Method for Solving Homogeneous Heat Diffusion Equation. IHJPAS. 2020, 33(2), 62-71, Doi: 10.30526/33.2.2427. 2. Al-Hawasy, J. A. A.; Mansour, N. F. 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