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Journal articles on the topic 'Runge-Kutta method of order four'

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Published: 2 June 2025
Last updated: 5 July 2025

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1

Hussain, Kasim A., and Waleed J. Hasan. "Improved Runge-Kutta Method for Oscillatory Problem Solution Using Trigonometric Fitting Approach." Ibn AL-Haitham Journal For Pure and Applied Sciences 36, no. 1 (2023): 345–54. http://dx.doi.org/10.30526/36.1.2963.

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This paper provides a four-stage Trigonometrically Fitted Improved Runge-Kutta (TFIRK4) method of four orders to solve oscillatory problems, which contains an oscillatory character in the solutions. Compared to the traditional Runge-Kutta method, the Improved Runge-Kutta (IRK) method is a natural two-step method requiring fewer steps. The suggested method extends the fourth-order Improved Runge-Kutta (IRK4) method with trigonometric calculations. This approach is intended to integrate problems with particular initial value problems (IVPs) using the set functions and for trigonometrically fitted. To improve the method's accuracy, the problem primary frequency is used. The novel method is more accurate than the conventional Runge-Kutta method and IRK4. Several test problems for the system of first-order ordinary differential equations carry out numerically to demonstrate the effectiveness of this approach. The computational studies show that the TFIRK4 approach is more efficient than the existing Runge-Kutta methods.
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2

Olatunji, Oladayo, and Adeyemi Akeju. "Comparative Analysis of Euler and Order Four Runge-Kutta Methods in Adams-Bashforth-Moulton Predictor-Corrector Method." International Journal of Mathematics, Statistics, and Computer Science 3 (January 2, 2025): 276–93. https://doi.org/10.59543/ijmscs.v3i.10471.

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This study conducts a comparative analysis of the Euler and Runge-Kutta 4 methods within the Adams Predictor-Corrector frameworkfor solving second-order ordinary differential equations (ODEs) andcoupled differential equations. The Euler method, a basic first-orderexplicit scheme, and the Runge-Kutta 4 method, a higher-order ex-plicit scheme, are widely utilized numerical methods for ODEs. How-ever, their effectiveness varies based on the problem’s characteristics.We specifically investigate these methods integrated into the AdamsPredictor-Corrector method, renowned for its stability and efficiencyin solving initial value problems. Through both numerical experimentsand theoretical analysis, we establish that the Runge-Kutta methoddemonstrates superior performance over the Euler method within thisframework. Additionally, we observe that the 4th order of the AdamsPredictor-Corrector method yields more accurate results than the 5thorder for second-order ODEs, while the 5th order performs better forcoupled ODEs. Overall, the Runge-Kutta method exhibits enhancedaccuracy and stability compared to the Euler method in the contextof the Adams Predictor-Corrector method.
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3

Ahmad, N. A., N. Senu, and F. Ismail. "Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs." Mathematical Problems in Engineering 2017 (2017): 1–11. http://dx.doi.org/10.1155/2017/1871278.

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A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods.
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Ahmad, S. Z., F. Ismail, N. Senu, and M. Suleiman. "Semi Implicit Hybrid Methods with Higher Order Dispersion for Solving Oscillatory Problems." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/136961.

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We constructed three two-step semi-implicit hybrid methods (SIHMs) for solving oscillatory second order ordinary differential equations (ODEs). The first two methods are three-stage fourth-order and three-stage fifth-order with dispersion order six and zero dissipation. The third is a four-stage fifth-order method with dispersion order eight and dissipation order five. Numerical results show that SIHMs are more accurate as compared to the existing hybrid methods, Runge-Kutta Nyström (RKN) and Runge-Kutta (RK) methods of the same order and Diagonally Implicit Runge-Kutta Nyström (DIRKN) method of the same stage. The intervals of absolute stability or periodicity of SIHM for ODE are also presented.
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Hussain, Kasim Abbas. "A new embedded improved runge-kutta approach to solve first order ordinary differential equation." Journal of Interdisciplinary Mathematics 28, no. 3-B (2025): 1247–54. https://doi.org/10.47974/jim-2214.

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This work presents a novel pair of embedded improved Runge-Kutta techniques for first-order ordinary differential equations (ODEs) numerical integration. Three and four are algebraic orders for this embedded scheme. The two-step approach shown here requires fewer steps per step than the current Runge-Kutta method. The newly embedded approach efficacy in contrast to the existing embedded Runge-Kutta technique is demonstrated numerically.
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6

Rabiei, Faranak, Fatin Abd Hamid, Nafsiah Md Lazim, Fudziah Ismail, and Zanariah Abdul Majid. "Numerical Solution of Volterra Integro-Differential Equations Using Improved Runge-Kutta Methods." Applied Mechanics and Materials 892 (June 2019): 193–99. http://dx.doi.org/10.4028/www.scientific.net/amm.892.193.

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In this paper, we proposed the numerical solution of Volterra integro-differential equations of the second kind using Improved Runge-Kutta method of order three and four with 2 stages and 4 stages, respectively. The improved Runge-kutta method is considered as two-step numerical method for solving the ordinary differential equation part and the integral operator in Volterra integro-differential equation is approximated using quadrature rule and Lagrange interpolation polynomials. To illustrate the efficiency of proposed methods, the test problems are carried out and the numerical results are compared with existing third and fourth order classical Runge-Kutta method with 3 and 4 stages, respectively. The numerical results showed that the Improved Runge-Kutta method by achieving the higher accuracy performed better results than existing methods.
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7

Fikri Haiqal Faturahman, F. Armawanto, D. A. Faradita, et al. "Optimalization of Neural Network Method on Harmonis Damped Swing." Jurnal Pendidikan Fisika Undiksha 13, no. 3 (2023): 370–81. https://doi.org/10.23887/jjpf.v13i3.70347.

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Numerical methods are used to solve the differential equations in the case of a damped pendulum system. In this paper, we aim to solve the equation of dumped pendulum motion using some numerical. Four numerical methods are decomposed, each the Euler method, fourth-order Runge-Kutte, Odeint provided in Python, and Neural Network scheme. All answers obtained are approximations or predictions that include errors. The error of those methods will be compared with the analytical solution of the case, known as the global error. The Odeint and fourth-order Runge-Kutta methods are more accurate than the other methods. The Odeint is built from the Runge-Kutta fourth-order method.The Neural Network method has less accuracy than Runge-Kutta, but the error obtained is still within acceptable tolerance limits.
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8

Lawal, Jibril* Ibrahim MaiHaja &. Ibrahim M. O. "SOLUTION OF CHOLERA CARRIER EPIDEMIC MODEL USING DIFFERENTIAL TRANSFORM METHOD (DTM)." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 7, no. 1 (2018): 242–49. https://doi.org/10.5281/zenodo.1136136.

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In this paper, Differential Transform Method (DTM) is applied to the deterministic mathematical model of cholera carrier epidemic. The model is transformed using DTM operational properties, hence, the power series of the model system is generated and also an approximate solution of the model was established. The accuracy of DTM is demonstrated against Fehlberg Runge-Kutta method with order four interpolant (RKF-45) numerical solution and it demonstrated high accuracy of the results. Plotted DTM solution is found to be in good agreement with the popular Runge-Kutta solution.In this paper, Differential Transform Method (DTM) is applied to the deterministic mathematical model of cholera carrier epidemic. The model is transformed using DTM operational properties, hence, the power series of the model system is generated and also an approximate solution of the model was established. The accuracy of DTM is demonstrated against Fehlberg Runge-Kutta method with order four interpolant (RKF-45) numerical solution and it demonstrated high accuracy of the results. Plotted DTM solution is found to be in good agreement with the popular Runge-Kutta solution.
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9

Ghazal, Zainab Khaled, and Kasim Abbas Hussain. "Solving Oscillating Problems Using Modifying Runge-Kutta Methods." Ibn AL- Haitham Journal For Pure and Applied Sciences 34, no. 4 (2021): 58–67. http://dx.doi.org/10.30526/34.4.2703.

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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.
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10

Rabiei, Faranak, Fudziah Ismail, Ali Ahmadian, and Soheil Salahshour. "Numerical Solution of Second-Order Fuzzy Differential Equation Using Improved Runge-Kutta Nystrom Method." Mathematical Problems in Engineering 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/803462.

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We develop the Fuzzy Improved Runge-Kutta Nystrom (FIRKN) method for solving second-order fuzzy differential equations (FDEs) based on the generalized concept of higher-order fuzzy differentiability. The scheme is two-step in nature and requires less number of stages which leads to less number of function evaluations in comparison with the existing Fuzzy Runge-Kutta Nystrom method. Therefore, the new method has a lower computational cost which effects the time consumption. We assume that the fuzzy function and its derivative are Hukuhara differentiable. FIRKN methods of orders three, four, and five are derived with two, three, and four stages, respectively. The numerical examples are given to illustrate the efficiency of the methods.
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11

S, Indrakumar* K. Kanagarajan. "RUNGE-KUTTA METHOD FOR FUZZY VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 5, no. 12 (2016): 666–78. https://doi.org/10.5281/zenodo.212020.

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12

Pratama, Muhammad Ridho, Fadli Handoyo, Shallu Fidhah Ariyanti, Delila Septiani Dwi Putri, and Michael Setiyanto Silambi. "Studi Projek Trajectories Partikel dalam Medan Magnet Non Homogen di ATLAS dengan Menggunakan Runge-Kutta-Nystrom dan Runge-Kutta-Nystrom Adaptif." Mitra Pilar: Jurnal Pendidikan, Inovasi, dan Terapan Teknologi 1, no. 1 (2022): 27–36. http://dx.doi.org/10.58797/pilar.0101.04.

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Abstract The problem that arises in the numerical method is the validation of the answers obtained from these calculations. Another phenomenon that can be analyzed using other numerical approaches was also discovered by the European Institute for Nuclear Research, CERN. four giant detectors record the energy-producing Higgs boson particles. One such detector is ATLAS. As for this research, a new adaptive Runge-Kutta-Nystrom method was developed for the STEP algorithm on the ATLAS detector. In solving problems regarding the determination of the second-order ordinary differential equation (2) on the path of particles in the non-homogeneous magnetic field of the ATLAS detector (A Toroidal LHC Apparatus). Two types of modified Runge-Kutta methods were used. This method is the adaptive Runge Kutta Nystrom method which is then compared with the usual Runge Kutta Nystrom method to find the function r(s) which is a 2nd-order differential function. The output table contains data for arc length (s), the radius of curvature (r), the gradient of the function R(dr/ds), and the step size (h).Runge-Kutta Nystrom Table Outputs. Adaptive Runge-Kutta Nystrom Graph Output. Runge-Kutta Nystrom Graph Output. Project Discussion. Based on Algorithm. Based on Time Consumption. Based on Cost-Computation. Based on the Step-Size Difference. Based on the time consumption, the Adaptive Runge-Kutta Nystrom algorithm requires a faster running time than the usual Runge-Kutta Nystrom algorithm. Adaptive Runge-Kutta Nystrom takes 1,095 seconds to run, and Runge Kutta Nystrom takes 2,135 seconds. If it is reviewed based on the computational side which is focused on memory efficiency, the Adaptive Runge-Kutta Nystrom method is better than Runge-Kutta Nystrom. Abstrak Permasalahan yang muncul pada metode numerik adalah validasi jawaban yang diperoleh dari perhitungan tersebut. Fenomena lain yang dapat dianalisis dengan menggunakan pendekatan numerik lainnya juga ditemukan oleh Lembaga Riset Nuklir Eropa, CERN. partikel Higgs boson yang menghasilkan energi direkam oleh empat detektor raksasa. Salah satu detektor tersebut adalah ATLAS. Adapun dalam penelitian ini dikembangkan metode Runge-Kutta-Nystrom adaptif yang baru untuk algoritma STEP pada detektor ATLAS Dalam menyelesaikan persoalan mengenai penentuan persamaan diferensial biasa orde 2 (dua) pada lintasan partikel dalam medan magnet non-homogen detektor ATLAS (A Toroidal LHC Apparatus) digunakan dua jenis metode Runge-Kutta yang dimodisikasi. Metode tersebut yaitu metode Runge Kutta Nystrom adaptif yang kemudian dibandingkan dengan metode Runge Kutta Nystrom biasa untuk mencari fungsi r(s) yang merupakan fungsi diferensial orde 2. Output Tabel yang berisikan data nilai panjang busur (s), jari-jari kelengkungan (r), gradien fungsi R(dr/ds), dan ukuran langkah (h). Output Tabel Runge-Kutta Nystrom. Output Grafik Runge-Kutta Nystrom Adaptif. Output Grafik Runge-Kutta Nystrom. Pembahasan Projek. Berdasarkan Algoritma. Berdasarkan Time Consumption. Berdasarkan Cost-Computation. Berdasarkan Perbedaan Step-Size.Bedasarkan sisi time consumption algoritma Runge-Kutta Nystrom Adaptif membutuhkan waktu running yang lebih cepat dibandingkan algoritma Runge-Kutta Nystrom biasa. Runge-Kutta Nystrom Adaptif membutuhkan waktu running selama 1.095 detik dan Runge Kutta Nystrom membutuhkan waktu selama 2.135 detik. Jika ditinjau berdasarkan sisi komputasi yang difokuskan pada efisiensi memori, maka dapat dikatakan bahwa metode Runge-Kutta Nystrom Adaptif lebih baik dari Runge-Kutta Nystrom.
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Kaur, Manpreet, Sangeet Kumar, and Jasdev Bhatti. "SOLUTION OF ORDINARY DIFFERENTIAL EQUATION vvi (u)=f(u,v,v',v'',v''') USING EIGHTH AND NINTH ORDER RUNGE-KUTTA TYPE METHOD." Malaysian Journal of Science 42, no. 2 (2023): 33–40. http://dx.doi.org/10.22452/mjs.vol42no2.5.

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The present paper presents the numerical conclusion to solve sixth order initial value ordinary differential equation (ODE). The concept of order conditions for three stage eighth order (RKSD8) & four stage ninth order Runge-Kutta methods (RKSD9) has been derived for finding global truncation error of differential equation The global and local truncated errors norms, zero stability of extended Runge-Kutta method (RK) is well defined and demonstrated with the help of an example.
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14

Journal, Baghdad Science. "Volterra Runge- Kutta Methods for Solving Nonlinear Volterra Integral Equations." Baghdad Science Journal 7, no. 3 (2010): 1270–74. http://dx.doi.org/10.21123/bsj.7.3.1270-1274.

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In this paper Volterra Runge-Kutta methods which include: method of order two and four will be applied to general nonlinear Volterra integral equations of the second kind. Moreover we study the convergent of the algorithms of Volterra Runge-Kutta methods. Finally, programs for each method are written in MATLAB language and a comparison between the two types has been made depending on the least square errors.
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15

Zhou, Naying, Hongxing Zhang, Wenfang Liu, and Xin Wu. "A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes." Astrophysical Journal 927, no. 2 (2022): 160. http://dx.doi.org/10.3847/1538-4357/ac497f.

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Abstract In recent publications, the construction of explicit symplectic integrators for Schwarzschild- and Kerr-type spacetimes is based on splitting and composition methods for numerical integrations of Hamiltonians or time-transformed Hamiltonians associated with these spacetimes. Such splittings are not unique but have various options. A Hamiltonian describing the motion of charged particles around the Schwarzschild black hole with an external magnetic field can be separated into three, four, and five explicitly integrable parts. It is shown through numerical tests of regular and chaotic orbits that the three-part splitting method is the best of the three Hamiltonian splitting methods in accuracy. In the three-part splitting, optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators exhibit the best accuracies. In fact, they are several orders of magnitude better than the fourth-order Yoshida algorithms for appropriate time steps. The first two algorithms have a small additional computational cost compared with the latter ones. Optimized sixth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators have no dramatic advantages over the optimized fourth-order ones in accuracy during long-term integrations due to roundoff errors. The idea of finding the integrators with the best performance is also suitable for Hamiltonians or time-transformed Hamiltonians of other curved spacetimes including Kerr-type spacetimes. When the numbers of explicitly integrable splitting sub-Hamiltonians are as small as possible, such splitting Hamiltonian methods would bring better accuracies. In this case, the optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström methods are worth recommending.
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Iavernaro, Felice, and Francesca Mazzia. "A Fourth Order Symplectic and Conjugate-Symplectic Extension of the Midpoint and Trapezoidal Methods." Mathematics 9, no. 10 (2021): 1103. http://dx.doi.org/10.3390/math9101103.

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The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method.
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17

Mingjing, Du, and Yulan Wang. "Some Novel Complex Dynamic Behaviors of a Class of Four-Dimensional Chaotic or Hyperchaotic Systems Based on a Meshless Collocation Method." Complexity 2019 (October 20, 2019): 1–15. http://dx.doi.org/10.1155/2019/5034025.

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In the field of complex systems, there is a need for better methods of knowledge discovery due to their nonlinear dynamics. The numerical simulation of chaotic or hyperchaotic system is mainly performed by the fourth-order Runge–Kutta method, and other methods are rarely reported in previous work. A new method, which divides the entire intervals into N equal subintervals based on a meshless collocation method, has been constructed in this paper. Some new complex dynamical behaviors are shown by using this new approach, and the results are in good agreement with those obtained by the fourth-order Runge–Kutta method.
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18

Fang, Xi, Dongbo Zhang, Xiaoyu Zhang, et al. "Chaos of flexible rotor system with critical speed in magnetic bearing based on the improved precise Runge–Kutta hybrid integration." Advances in Mechanical Engineering 10, no. 9 (2018): 168781401880085. http://dx.doi.org/10.1177/1687814018800859.

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Magnetic rotor-bearing system has drawn great attention because of its several advantages compared to existent rotor-bearing system, and explicit Runge–Kutta method has achieved good results in solving dynamic equation. However, research on flexible rotor of magnetic bearing is relatively less. Moreover, explicit Runge–Kutta needs a smaller integral step to ensure the stability of the calculation. In this article, we propose a nonlinear dynamic analysis of flexible rotor of active magnetic bearing established by using the finite element method. The precise Runge–Kutta hybrid integration method and the largest Lyapunov exponent are used to analyze the chaos of the rotor system at the first- and second-order critical speed of the rotor. Experiment on chaos analysis has shown that compared with the explicit Runge–Kutta method, the precise Runge–Kutta hybrid integration method can improve the convergence step of calculation significantly while avoiding iterative solution and maintain high accuracy which is four times the increase of the integral step.
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Rajakumari, P. Evangelin Diana, and R. Gethsi Sharmila. "Runge-Kutta Method Based on the Linear Combination of Arithmetic Mean, Geometric Mean, and Centroidal Mean: A Numerical Solution for the Hybrid Fuzzy Differential Equation." Indian Journal Of Science And Technology 17, no. 18 (2024): 1860–67. http://dx.doi.org/10.17485/ijst/v17i18.75.

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Objectives: To find the absolute error of the Hybrid fuzzy differential equation with triangular fuzzy number since its membership function has a triangular form. Methods: The third order runge-kuttta method based on the linear combination of Arithmetic mean, Geometric mean and Centroidal mean is introduced to solve Hybrid fuzzy differential equation. Seikkala’s derivatives are considered, and Numerical example is given to demonstrate the effectiveness of the proposed method. Findings: The comparative study have been done between the proposed method and the existing third order runge-kutta method based on Arithmetic mean. The proposed method gives better error tolerance than the third order runge-kutta method based on Arithmetic mean. Novelty: In this study a new formula has been developed by combining three means Arithmetic Mean, Geometric Mean and Centroidal Mean using Khattri’s formula and it is solved by the third order runge-kutta method for the first order hybrid fuzzy differential equation. The error tolerance has been calculated for the proposed method and the Arithmetic mean and it is seen that the error tolerance is better for the proposed method. Keywords: Hybrid fuzzy differential equations, Triangular fuzzy number, Seikkala’s derivative, Third order runge-kutta method, Initial value problem
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20

Senu, Norazak, Mohamed Suleiman, Fudziah Ismail, and Norihan Md Arifin. "New 4(3) Pairs Diagonally Implicit Runge-Kutta-Nyström Method for Periodic IVPs." Discrete Dynamics in Nature and Society 2012 (2012): 1–20. http://dx.doi.org/10.1155/2012/324989.

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New 4(3) pairs Diagonally Implicit Runge-Kutta-Nyström (DIRKN) methods with reduced phase-lag are developed for the integration of initial value problems for second-order ordinary differential equations possessing oscillating solutions. Two DIRKN pairs which are three- and four-stage with high order of dispersion embedded with the third-order formula for the estimation of the local truncation error. These new methods are more efficient when compared with current methods of similar type and with the L-stable Runge-Kutta pair derived by Butcher and Chen (2000) for the numerical integration of second-order differential equations with periodic solutions.
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Vega, Carlos A., and Francisco Arias. "Numerical Simulations of a Polydisperse Sedimentation Model by Using Spectral WENO Method with Adaptive Multiresolution." International Journal of Computational Methods 13, no. 06 (2016): 1650037. http://dx.doi.org/10.1142/s0219876216500377.

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In this work, we apply adaptive multiresolution (Harten’s approach) characteristic-wise fifth-order Weighted Essentially Non-Oscillatory (WENO) for computing the numerical solution of a polydisperse sedimentation model, namely, the Höfler and Schwarzer model. In comparison to other related works, time discretization is carried out with the ten-stage fourth-order strong stability preserving Runge–Kutta method which is more efficient than the widely used optimal third-order TVD Runge–Kutta method. Numerical results with errors, convergence rates and CPU times are included for four and 11 species.
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Ghawadri, Nizam, Norazak Senu, Firas Adel Fawzi, Fudziah Ismail, and Zarina Ibrahim. "Diagonally Implicit Runge–Kutta Type Method for Directly Solving Special Fourth-Order Ordinary Differential Equations with Ill-Posed Problem of a Beam on Elastic Foundation." Algorithms 12, no. 1 (2018): 10. http://dx.doi.org/10.3390/a12010010.

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In this study, fifth-order and sixth-order diagonally implicit Runge–Kutta type (DIRKT) techniques for solving fourth-order ordinary differential equations (ODEs) are derived which are denoted as DIRKT5 and DIRKT6, respectively. The first method has three and the another one has four identical nonzero diagonal elements. A set of test problems are applied to validate the methods and numerical results showed that the proposed methods are more efficient in terms of accuracy and number of function evaluations compared to the existing implicit Runge–Kutta (RK) methods.
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23

Chou, Lin-Yi, and P. W. Sharp. "On order 5 symplectic explicit Runge-Kutta Nyström methods." Journal of Applied Mathematics and Decision Sciences 4, no. 2 (2000): 143–50. http://dx.doi.org/10.1155/s1173912600000109.

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Order five symplectic explicit Runge-Kutta Nyström methods of five stages are known to exist. However, these methods do not have free parameters with which to minimise the principal error coefficients. By adding one derivative evaluation per step, to give either a six-stage non-FSAL family or a seven-stage FSAL family of methods, two free parameters become available for the minimisation. This raises the possibility of improving the efficiency of order five methods despite the extra cost of taking a step.We perform a minimisation of the two families to obtain an optimal method and then compare its numerical performance with published methods of orders four to seven. These comparisons along with those based on the principal error coefficients show the new method is significantly more efficient than the five-stage, order five methods. The numerical comparisons also suggest the new methods can be more efficient than published methods of other orders.
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24

Qasim Abd Ali Tayyeh. "Some Application for a Fuzzy Differential Equation and Solve by Runge-Kutta Method." Bilangan : Jurnal Ilmiah Matematika, Kebumian dan Angkasa 2, no. 4 (2024): 248–61. http://dx.doi.org/10.62383/bilangan.v2i4.206.

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In this article, the starting condition was defined using a fuzzy initial value problem (IVP). Additionally, we discussed various methods for solving fuzzy differential equations, including the modified two-step Simpson method and Runge-Kutta of orders (two, three, four, five, and six). For each method, we provided a numerical example and the known convergence rates of the solutions. Then we discussed the comparison of the solutions of all methods, using computer software to offer rough solutions for the Runge Kutta method. And take some application solve by Runge-Kutta in physics and medical
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Arar, Nouria, Leila Ait Kaki, and Abdellatif Ben Makhlouf. "Highly Efficacious Sixth-Order Compact Approach with Nonclassical Boundary Specifications for the Heat Equation." Mathematical Problems in Engineering 2022 (December 30, 2022): 1–13. http://dx.doi.org/10.1155/2022/8224959.

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This paper suggests an accurate numerical method based on a sixth-order compact difference scheme and explicit fourth-order Runge–Kutta approach for the heat equation with nonclassical boundary conditions (NCBC). According to this approach, the partial differential equation which represents the heat equation is transformed into several ordinary differential equations. The system of ordinary differential equations that are dependent on time is then solved using a fourth-order Runge–Kutta method. This study deals with four test problems in order to provide evidence for the accuracy of the employed method. After that, a comparison is done between numerical solutions obtained by the proposed method and the analytical solutions as well as the numerical solutions available in the literature. The proposed technique yields more accurate results than the existing numerical methods.
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Olaniyan, Adegoke, Moses Akanbi, Ashiribo Wusu, and Kazeem Shonibare. "A four-stage multiderivative explicit Runge-Kutta method for the solution of first order ordinary differential equations." Annals of Mathematics and Computer Science 20 (January 3, 2024): 72–81. http://dx.doi.org/10.56947/amcs.v20.224.

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The development of numerical methods for the solution of initial value problems in ordinary differential equation have turned out to be a very rapid research area in recent decades due to the difficulties encountered in finding solutions to some mathematical models composed into differential equations from real life situations. Researchers have in recent times, used higher derivatives in the derivation of numerical methods to produce totally new ways of solving these equations. In this article, a new Runge-Kutta type methods with reduced number of function evaluations in the increment function is constructed, analyzed and implemented. This proposed method border on the use of higher derivatives up to the second derivative in the ki terms of Runge-Kutta method in order to achieve a higher order of accuracy. The qualitative features: local truncation error, consistency, convergence and stability of the new method were investigated and established. Numerical examples were also performed on some initial value problems to confirm the accuracy of the new method and compared with some existing methods of which the numerical results show that the new method competes favorably.
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Et al., Hussain. "A New Two Derivative FSAL Runge-Kutta Method of Order Five in Four Stages." Baghdad Science Journal 17, no. 1 (2020): 0166. http://dx.doi.org/10.21123/bsj.2020.17.1.0166.

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A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods.
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Che, Yuzhang, Chungang Chen, Feng Xiao, Xingliang Li, and Xueshun Shen. "A Two-Stage Fourth-Order Multimoment Global Shallow-Water Model on the Cubed Sphere." Monthly Weather Review 148, no. 10 (2020): 4267–79. http://dx.doi.org/10.1175/mwr-d-20-0004.1.

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AbstractA new multimoment global shallow-water model on the cubed sphere is proposed by adopting a two-stage fourth-order Runge–Kutta time integration. Through calculating the values of predicted variables at half time step t = tn + (1/2)Δt by a second-order formulation, a fourth-order scheme can be derived using only two stages within one time step. This time integration method is implemented in our multimoment global shallow-water model to build and validate a new and more efficient numerical integration framework for dynamical cores. As the key task, the numerical formulation for evaluating the derivatives in time has been developed through the Cauchy–Kowalewski procedure and the spatial discretization of the multimoment finite-volume method, which ensures fourth-order accuracy in both time and space. Several major benchmark tests are used to verify the proposed numerical framework in comparison with the existing four-stage fourth-order Runge–Kutta method, which is based on the method of lines framework. The two-stage fourth-order scheme saves about 30% of the computational cost in comparison with the four-stage Runge–Kutta scheme for global advection and shallow-water models. The proposed two-stage fourth-order framework offers a new option to develop high-performance time marching strategy of practical significance in dynamical cores for atmospheric and oceanic models.
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Yang, Baonan, Zhen Wang, Huaigu Tian, and Jindong Liu. "Symplectic Dynamics and Simultaneous Resonance Analysis of Memristor Circuit Based on Its van der Pol Oscillator." Symmetry 14, no. 6 (2022): 1251. http://dx.doi.org/10.3390/sym14061251.

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A non-autonomous memristor circuit based on van der Pol oscillator with double periodically forcing term is presented and discussed. Firstly, the differences of the van der Pol oscillation of memristor model between Euler method and symplectic Euler method, four-order Runge–Kutta method (RK4) and four-order symplectic Runge–Kutta–Nyström method (SRKN4), symplectic Euler method and RK4 method, and symplectic Euler method and SRKN4 method in preserving structure are compared from theoretical and numerical simulations, the symmetry and structure preserving and numerical stability of symplectic scheme are demonstrated. Moreover, the analytic solution of the primary and subharmonic simultaneous resonance of this system is obtained by using the multi-scale method. Finally, based on the resonance relation of the system, the chaotic dynamics behaviors with different parameters are studied.
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Kaur, Manpreet, Jasdev Bhatti, Sangeet Kumar, and Srinivasarao Thota. "Explicit Runge-Kutta Method for Evaluating Ordinary Differential Equations of type v^vi = f(u, v, v′)." WSEAS TRANSACTIONS ON MATHEMATICS 23 (March 22, 2024): 167–75. http://dx.doi.org/10.37394/23206.2024.23.20.

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The initiative of this paper is to present the Runge Kutta Type technique for the development of mathematical solutions to the problems concerning to ordinary differential equation of order six of structure v vi = f(u, v, v′ ) denoted as RKSD with initial conditions. The three and four stage Runge-Kutta methods with order conditions up to order seven (RKSD7) have been designed to evaluate global and local truncated errors for the ordinary differential equation of order six. The framework and evaluation of equations with their results are well established to obtain the effectiveness of RK method towards implicit function satisfying the required initial conditions and for obtaining zero-stability of RKSD7 in terms of their accuracy with maximum precision under minimal processing.
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31

MOHAMMADPOURFARD, M., and M. FALLAH. "OPTIMIZED FREE ENERGY-BASED LATTICE BOLTZMANN METHOD FOR MODELING MICRO DROP DYNAMICS." International Journal of Computational Methods 10, no. 03 (2013): 1350006. http://dx.doi.org/10.1142/s0219876213500060.

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The free energy-based lattice Boltzmann (FEB-LB) model is a very powerful method and is fully consistent with Maxwell's equal-area construction, especially for low temperature conditions. Drawbacks of this model are its massive amounts of calculations and large computation time (i.e., iteration number), therefore it is often executed on supercomputers or parallel computing systems. In this paper, the three-dimensional FEB-LB model has been optimized using the Runge-Kutta fourth-order method. It is shown that using the fourth-order Runge–Kutta method, the computation time is decreased about four times. In addition, in the present work, using the optimized FEB-LB model, the effects of surface tension and interface width on spurious velocities in the liquid–gas interface have been investigated.
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32

Wu, Zhu Li. "Analysis on Differential Equation of Decision Model Based on Matlab Simulation." Applied Mechanics and Materials 602-605 (August 2014): 3301–5. http://dx.doi.org/10.4028/www.scientific.net/amm.602-605.3301.

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This paper introduces Runge-Kutta method into the game decision model of advertising differential equation, and establishes the two-order Runge-Kutta and the four-order Runge-Kutta differential equation models to study the impact of new fashion product on the sales and the impact of supply chain on the overall performance. In order to verify the validity and reliability of the model, this paper uses the MATLAB software to design the algorithm of advertising promotion differential equation model, and gets the accounting system of three types of advertising sales, and uses this system to statistics the sales. From the results we can see, the popular goods and seasonal goods have the largest effect on the new development fashion goods sales in three types of advertising, so it should increase the advertising delivery in popular goods and seasonal goods to ensure the sales of new fashionable goods.
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33

Salih, M., F. Ismail, and N. Senu. "PHASE FITTED CLASSICAL RUNGE-KUTTA METHOD OF ORDER FOUR FOR SOLVING OSCILLATORY PROBLEMS." Far East Journal of Mathematical Sciences (FJMS) 96, no. 5 (2015): 615–28. http://dx.doi.org/10.17654/fjmsmar2015_615_628.

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34

J. Hasan, Waleed, and Kasim A. Hussain. "Fifth Order Improved Runge-Kutta Nystrom Method Using Trigonometrically-Fitting for Solving Oscillatory Problems." Al-Nahrain Journal of Science 25, no. 4 (2022): 63–67. http://dx.doi.org/10.22401/anjs.25.4.11.

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In this paper, the Trigonometrically Fitted Improved Runge-KuttaNystrom method is proposed as a novel method with four stages and fifth order for solving oscillatory problems. This method is intended to integrate second-order initial value problems using the trigonometrically fitting approach. To increase the method'saccuracy, the principal frequency of the problem푤∈ℝ, is used. It is discovered that the new method is more precise when compared with the other existing Runge-Kutta Nystrom and IRKN5 methods. To show how well the TFIRKN5 method works, test problems for second-order ordinary differential equations (ODEs) are solved. The numerical outcomes show that the novel approach outperforms methods that have already been published.
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SIMOS, T. E. "AN EMBEDDED RUNGE–KUTTA METHOD WITH PHASE-LAG OF ORDER INFINITY FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION." International Journal of Modern Physics C 11, no. 06 (2000): 1115–33. http://dx.doi.org/10.1142/s0129183100000973.

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An embedded Runge–Kutta method with phase-lag of order infinity for the numerical integration of Schrödinger equation is developed in this paper. The methods of the embedded scheme have algebraic orders five and four. Theoretical and numerical results obtained for radial Schrödinger equation and for coupled differential equations show the efficiency of the new methods.
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36

Husin, Nurain Zulaikha, and Muhammad Zaini Ahmad. "Hybridization of the shooting and Runge-Kutta Cash-Karp methods for solving Fuzzy Boundary Value Problems." AIMS Mathematics 9, no. 11 (2024): 31806–47. http://dx.doi.org/10.3934/math.20241529.

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<p>Fuzzy Differential Equations (FDEs) have attracted great interest among researchers. These FDEs have been used to develop a mathematical model for everyday life problems. In this study, we propose a solution method for a second-order Fuzzy Boundary Value Problem (FBVP). Four systems of FBVPs were developed based on the generalized fuzzy derivative. The second-order FBVP for each system was divided into two parts: Fuzzy non-homogeneous and fuzzy homogeneous equations. Using the shooting method, these two equations were then reduced to first-order FDEs. By implementing the Fuzzy Runge-Kutta Cash-Karp of the fourth-order method (FRKCK4), the approximate solution was compared with the analytical solution and the solution from the Fuzzy Runge-Kutta of the fourth-order method (FRK4).</p>
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Lai, Tao, Ting-Hua Yi, Hong-Nan Li, and Xing Fu. "An Explicit Fourth-Order Runge–Kutta Method for Dynamic Force Identification." International Journal of Structural Stability and Dynamics 17, no. 10 (2017): 1750120. http://dx.doi.org/10.1142/s0219455417501206.

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This paper presents a new technique for input reconstruction based on the explicit fourth-order Runge–Kutta (RK4) method. First, the state-space representation of the dynamic system is discretized by the explicit RK4 method under the assumption of linear interpolation for the dynamic load, leading to a recurrence equation between the current state and the previous state. Then, the mapping from the sequences of input to output is established through the recursive operation of the system equation and observation equation. Finally, the stabilized force information is recovered using the Tikhonov regularization method. This approach makes use of the good stability and high precision of the RK4 method; in addition, the computational efficiency is enhanced by avoiding the computation of the inverse stiffness matrix. The proposed method is numerically illustrated and validated with various excitations on a simple four-story shear building and a more complicated 2D truss structure, along with a detailed parametric study. The simulation studies show that the external loads can be reconstructed with high efficiency and accuracy under a low noise environment.
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38

Moo, K. W., N. Senu, F. Ismail, and M. Suleiman. "New Phase-Fitted and Amplification-Fitted Fourth-Order and Fifth-Order Runge-Kutta-Nyström Methods for Oscillatory Problems." Abstract and Applied Analysis 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/939367.

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Two new Runge-Kutta-Nyström (RKN) methods are constructed for solving second-order differential equations with oscillatory solutions. These two new methods are constructed based on two existing RKN methods. Firstly, a three-stage fourth-order Garcia’s RKN method. Another method is Hairer’s RKN method of four-stage fifth-order. Both new derived methods have two variable coefficients with phase-lag of order infinity and zero amplification error (zero dissipative). Numerical tests are performed and the results show that the new methods are more accurate than the other methods in the literature.
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MASET, STEFANO, LUCIO TORELLI, and ROSSANA VERMIGLIO. "RUNGE–KUTTA METHODS FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS." Mathematical Models and Methods in Applied Sciences 15, no. 08 (2005): 1203–51. http://dx.doi.org/10.1142/s0218202505000716.

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We introduce Runge–Kutta (RK) methods for Retarded Functional Differential Equations (RFDEs). With respect to RK methods (A, b, c) for Ordinary Differential Equations the weights vector b ∈ ℝs and the coefficients matrix A ∈ ℝs×s are replaced by ℝs-valued and ℝs×s-valued polynomial functions b(·) and A(·) respectively. Such methods for RFDEs are different from Continuous RK (CRK) methods where only the weights vector is replaced by a polynomial function. We develop order conditions and construct explicit methods up to the convergence order four.
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40

Lee, Khai Chien, Norazak Senu, Ali Ahmadian, and Siti Nur Iqmal Ibrahim. "On Two-Derivative Runge–Kutta Type Methods for Solving u‴ = f(x,u(x)) with Application to Thin Film Flow Problem." Symmetry 12, no. 6 (2020): 924. http://dx.doi.org/10.3390/sym12060924.

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A class of explicit Runge–Kutta type methods with the involvement of fourth derivative, denoted as two-derivative Runge–Kutta type (TDRKT) methods, are proposed and investigated for solving a special class of third-order ordinary differential equations in the form u ‴ ( x ) = f ( x , u ( x ) ) . In this paper, two stages with algebraic order four and three stages with algebraic order five are presented. The derivation of TDRKT methods involves single third derivative and multiple evaluations of fourth derivative for every step. Stability property of the methods are analysed. Accuracy and efficiency of the new methods are exhibited through numerical experiments.
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41

Jia, Xinyue, Chang Liu, and Wenan Jiang. "LIE DERIVATIVE DISCRETIZATION SCHEME FOR SOLVING VARIABLE MASS SYSTEMS." Journal of the Serbian Society for Computational Mechanics 18, no. 2 (2024): 33–47. https://doi.org/10.24874/jsscm.2024.18.02.02.

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Lie derivative plays a key role in mathematics and physics. In particular, the Lie derivative discretization scheme has been implemented for a relatively large time step in order to quickly solve the numerical solution of the system. In this paper, an algorithm of the Lie derivative discretization scheme is applied to variable mass systems. Four different types of variable mass systems are employed to study the numerical solutions, and the calculated results are consistent with those obtained by the fourth-order Runge-Kutta method. Computational experiments demonstrate the success of the proposed method on variable mass systems. Moreover, the algorithm of Lie derivative discretization is shown to have superior computational efficiency and larger time steps compared to the Runge-Kutta algorithm.
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42

Fang, Yonglei, Qinghong Li, Qinghe Ming, and Kaimin Wang. "A New Optimized Runge-Kutta Pair for the Numerical Solution of the Radial Schrödinger Equation." Abstract and Applied Analysis 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/641236.

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A new embedded pair of explicit modified Runge-Kutta (RK) methods for the numerical integration of the radial Schrödinger equation is presented. The two RK methods in the pair have algebraic orders five and four, respectively. The two methods of the embedded pair are derived by nullifying the phase lag, the first derivative of the phase lag of the fifth-order method, and the phase lag of the fourth-order method. Nu merical experiments show the efficiency and robustness of our new methods compared with some well-known integrators in the literature.
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Demba, Musa Ahmed, Norazak Senu, Higinio Ramos, and Wiboonsak Watthayu. "Development of an Efficient Diagonally Implicit Runge–Kutta–Nyström 5(4) Pair for Special Second Order IVPs." Axioms 11, no. 10 (2022): 565. http://dx.doi.org/10.3390/axioms11100565.

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In this work, a new pair of diagonally implicit Runge–Kutta–Nyström methods with four stages is constructed. The proposed method has been derived to solve initial value problems of special second-order ordinary-differential equations. The principal local truncation error of the new method is obtained, and the main characteristics of the new method are analyzed. Some numerical experiments are performed, which demonstrate the robustness and efficiency of the new embedded pair.
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Olemskoy, Igor V., Alexey S. Eremin, and Oksana S. Firyulina. "A nine-parametric family of embedded methods of sixth order." Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes 19, no. 4 (2023): 449–68. http://dx.doi.org/10.21638/11701/spbu10.2023.403.

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In the paper an effective explicit Runge—Kutta type method of the sixth order with an embedded error estimator of order four is presented. The method is applied to the systems that can be structurally partitioned into three subsystems. Its computational scheme effectively uses the structural properties. However this leads to much larger systems of order conditions. These nonlinear conditions and the algorithm of finding a solution with nine free parameters are presented. A certain computational scheme is written down and a numerical comparison to Dormand—Prince pairs of orders 5 and 6 is performed.
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45

Sabir, Zulqurnain, Juan L. G. Guirao, Tareq Saeed, and Fevzi Erdoğan. "Design of a Novel Second-Order Prediction Differential Model Solved by Using Adams and Explicit Runge–Kutta Numerical Methods." Mathematical Problems in Engineering 2020 (July 10, 2020): 1–7. http://dx.doi.org/10.1155/2020/9704968.

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In this study, a novel second-order prediction differential model is designed, and numerical solutions of this novel model are presented using the integrated strength of the Adams and explicit Runge–Kutta schemes. The idea of the present study comes to the mind to see the importance of delay differential equations. For verification of the novel designed model, four different examples of the designed model are numerically solved by applying the Adams and explicit Runge–Kutta schemes. These obtained numerical results have been compared with the exact solutions of each example that indicate the performance and exactness of the designed model. Moreover, the results of the designed model have been presented numerically and graphically.
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HANSEN, JAKOB, ALEXEI KHOKHLOV, and IGOR NOVIKOV. "PROPERTIES OF FOUR NUMERICAL SCHEMES APPLIED TO A NONLINEAR SCALAR WAVE EQUATION WITH A GR-TYPE NONLINEARITY." International Journal of Modern Physics D 13, no. 05 (2004): 961–82. http://dx.doi.org/10.1142/s021827180400502x.

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We study stability, dispersion and dissipation properties of four numerical schemes (Itera-tive Crank–Nicolson, 3rd and 4th order Runge–Kutta and Courant–Fredrichs–Levy Nonlinear). By use of a Von Neumann analysis we study the schemes applied to a scalar linear wave equation as well as a scalar nonlinear wave equation with a type of nonlinearity present in GR-equations. Numerical testing is done to verify analytic results. We find that the method of lines (MOL) schemes are the most dispersive and dissipative schemes. The Courant–Fredrichs–Levy Nonlinear (CFLN) scheme is most accurate and least dispersive and dissipative, but the absence of dissipation at Nyquist frequency, if fact, puts it at a disadvantage in numerical simulation. Overall, the 4th order Runge–Kutta scheme, which has the least amount of dissipation among the MOL schemes, seems to be the most suitable compromise between the overall accuracy and damping at short wavelengths.
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47

Azizan, Farah Liyana, Saratha Sathasivam, Majid Khan Majahar Ali, Hazleen Marleesa A. H. Hafeez, and Nurizzati Jaafar. "Forecasting Modelling of Adolescent Pregnancy Crisis in Malaysia Using Runge-Kutta Fourth Order Method." EDUCATUM Journal of Science, Mathematics and Technology 10, no. 2 (2023): 11–18. http://dx.doi.org/10.37134/ejsmt.vol10.2.2.2023.

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Despite the significant dangers that countries suffer when adolescent pregnancy is neglected, this major problem usually receives little attention. In recent decades, the number of teenage girls who get pregnant has increased worldwide. Therefore, this issue has become many countries' main public health concern. This paper aims to use MATLAB to estimate Malaysian teen pregnancy data for the next few years and to find missing data for a particular year. Since the Runge-Kutta is used to forecast the prevalence of adolescent pregnancy in Malaysia, the comparison reveals that the two sets of data are distinct. Therefore, the order four Runge-Kutta method is insufficient to approximate incomplete data for adolescent pregnancy. Although the results show that adolescent pregnancy has decreased over time, we must continue to treat this issue seriously. It is necessary to educate teenagers about sex education and safe pregnancy. Educating the young about the risk of having pregnancies in their early years is also necessary. Additionally, it is necessary to provide special consideration to those who did not grow up in a good environment and belong to a family unit that does not function in a healthy way according to social norms.
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Khan, Muhammad Altaf, S. F. Saddiq, Saeed Islam, Ilyas Khan, and Dennis Ling Chuan Ching. "Epidemic Model of Leptospirosis Containing Fractional Order." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/317201.

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We study an epidemic model of leptospirosis in fractional order numerically. The multistep generalized differential transform method is applied to find the accurate approximate solution of the epidemic model of leptospirosis disease in fractional order. A unique positive solution for the epidemic model in fractional order is presented. For the integer case derivative, the approximate solution of MGDTM is compared with the Runge-Kutta order four scheme. The numerical results are presented for the justification purpose.
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Indrakumar, S., та K. Kanagarajan. "Runge-Kutta method of order four for solving fuzzy delay differential equations under generalized differentiability". Journal of Applied Nonlinear Dynamics 7, № 2 (2018): 131–46. http://dx.doi.org/10.5890/jand.2018.06.003.

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50

N. P., Okafor, Ogunlusi T. A., Ogunwe F. T., and Ayobami A. I. "Dynamic Analysis of an Exponentially Decaying Foundation on the Response of Non-Uniform Damped Rayleigh Beam under Harmonic Moving Load with General Boundary Conditions." Asian Research Journal of Mathematics 21, no. 1 (2025): 35–69. https://doi.org/10.9734/arjom/2025/v21i1882.

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This study investigates the dynamic response of a non-uniform damped Rayleigh beam on an exponentially decaying foundation subjected to a harmonic moving load with general boundary conditions. The governing equation, a fourth-order non-homogeneous partial differential equation with variable coefficients, is discretized using the Generalized Galerkin Method. Two cases are examined: moving force and moving mass. Closedform solutions are obtained for the moving force case using Laplace transform in conjuction with convolution theorem. For the moving mass case, the Struble asymptotic method cannot simplify the equation for the moving mass case due to the variable load magnitude, and thus, Runge-Kutta method of order four (RK4) is employed to obtain a numerical solution. Analytical and numerical solutions are compared for validation of accuracy of the Runge-kutta scheme and found compared favourably. The effects of some key structural parameters on dynamic behavior are examined, and resonance conditions are established.
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