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Journal articles on the topic 'RUNGE-KUTTA FOURTH ORDER METHOD'

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Published: 16 January 2024

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1

Suryani, Irma, Wartono Wartono, and Yuslenita Muda. "Modification of Fourth order Runge-Kutta Method for Kutta Form With Geometric Means." Kubik: Jurnal Publikasi Ilmiah Matematika 4, no. 2 (February 25, 2020): 221–30. http://dx.doi.org/10.15575/kubik.v4i2.6425.

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This paper discuss how to modified Fourth order Runge-Kutta Kutta method based on the geometric mean. Then we have parameters and however by re-comparing the Taylor series expansion of and up to the 4th order. For make error term re-compering of the Taylor series expansion of and up to the 5th order. In the error term an make substitution for the values of and into the Taylor seriese expansion up to the 5th order. So that we have error term modified Fourth Order Runge-Kutta Kutta based on the geometric mean. Modified Fourth Order Runge-Kutta Kutta based on the geometric mean that usually used to solved ordinary differential equations.

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2

Trifina, Leonora L. R., Ali Warsito, Laura A. S. Lapono, and Andreas Ch Louk. "VISUALISASI FENOMENA HARMONIS DAN CHAOS PADA GETARAN TERGANDENG BERBASIS KOMPUTASI NUMERIK RUNGE KUTTA." Jurnal Fisika : Fisika Sains dan Aplikasinya 8, no. 1 (April 27, 2023): 11–20. http://dx.doi.org/10.35508/fisa.v8i1.11817.

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Research has been carried out on the visualization of harmonic and chaos phenomenont on coupled vibration physcal case using the Runge Kutta numerical computation method with the aim of applying the first to fourth order Runge Kutta computation method to obtain a second order differential equation solution on coupled vibration system, calculating the displacement value of objects using computation method Runge Kutta order first to fourth, obtained a graph of the displacement of objects againts time in case of coupled vibration for harmonic and chaos states at certain step width values and compare the convergence of the Runge Kutta method from first to fourth order with the special analytical method. The solution of coupled vibration equation which is classified as a second order differential equation was quite difficulted to solve analytically, so the Runge Kutta computation method was used to solve it as an alternative solution. The results of the research showed that the harmonic state of the system was obtained when the displacement graph showed the motion of each pendulum which was constant with the pendulum displacement position with respect to time in the form of a sinusoidal graph at a value of C1 = 40 N/m, C2 = 30 N/m, C = 10 N/m, C = 0 N/m and the chaotic state was represented by a graph of the displacement of the pendulum with respect to time with an irregular pattern. In this case, it was found that the fourth order Runge Kutta method converged faster than the first to third order Runge Kutta method with the best results obtained at a step width value of 0,001. The fourth order Runge Kutta method also has a smaller approximation average error value from first to third order Runge Kutta method was on the fourth order Runge Kutta method and the avarage error values are , and on the Runge Kutta method of first to third order.

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3

Husin, Nurain Zulaikha, Muhammad Zaini Ahmad, and Mohd Kamalrulzaman Md Akhir. "Incorporating Fuzziness in the Traditional Runge–Kutta Cash–Karp Method and Its Applications to Solve Autonomous and Non-Autonomous Fuzzy Differential Equations." Mathematics 10, no. 24 (December 8, 2022): 4659. http://dx.doi.org/10.3390/math10244659.

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The study of the fuzzy differential equation is a topic that researchers are interested in these days. By modelling, this fuzzy differential equation can be used to resolve issues in the real world. However, finding an analytical solution to this fuzzy differential equation is challenging. Thus, this study aims to present the fuzziness in the traditional Runge–Kutta Cash–Karp of the fourth-order method to solve the first-order fuzzy differential equation. Later, this method is referred to as the fuzzy Runge–Kutta Cash–Karp of the fourth-order method. There are two types of fuzzy differential equations to be solved: autonomous and non-autonomous fuzzy differential equations. This fuzzy differential equation is divided into the (i) and (ii)–differentiability on the basis of the characterization theorem. The convergence analysis of the fuzzy Runge–Kutta Cash–Karp of the fourth-order method is also presented. By implementing the fuzzy Runge–Kutta Cash–Karp of the fourth-order method, the approximate solution is compared with the analytical and numerical solutions obtained from the fuzzy Runge–Kutta of the fourth-order method. The results demonstrated that the approximate solutions of the proposed method are accurate with an analytical solution, when compared with the solutions of the fuzzy Runge–Kutta of the fourth-order method.

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4

Rijoly, Monalisa E., and Francis Yunito Rumlawang. "Penyelesaian Numerik Persamaan Diferensial Orde Dua Dengan Metode Runge-Kutta Orde Empat Pada Rangkaian Listrik Seri LC." Tensor: Pure and Applied Mathematics Journal 1, no. 1 (May 28, 2020): 7–14. http://dx.doi.org/10.30598/tensorvol1iss1pp7-14.

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One alternative to solve second order differential equations by numerical methods, specificallynon-liner differential equations is the Runge-Kutta fourth order method. The Runge-Kutta fourth ordermethod is a numerical method that has high degree of precision and accuracy when compared to othernumerical methods. In this paper we will discuss the numerical solution of second order differentialequations on LC series circuit problem using the Runge-Kutta fourth order method. The numericalsolution generated by the computational calculation using the MATLAB program, the strong current andcharge are obtaind from t = 0 and t =0,5 second and different step size values

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5

Hussain, Kasim, Fudziah Ismail, and Norazak Senu. "Runge-Kutta Type Methods for Directly Solving Special Fourth-Order Ordinary Differential Equations." Mathematical Problems in Engineering 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/893763.

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A Runge-Kutta type method for directly solving special fourth-order ordinary differential equations (ODEs) which is denoted by RKFD method is constructed. The order conditions of RKFD method up to order five are derived; based on the order conditions, three-stage fourth- and fifth-order Runge-Kutta type methods are constructed. Zero-stability of the RKFD method is proven. Numerical results obtained are compared with the existing Runge-Kutta methods in the scientific literature after reducing the problems into a system of first-order ODEs and solving them. Numerical results are presented to illustrate the robustness and competency of the new methods in terms of accuracy and number of function evaluations.

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6

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 (January 20, 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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7

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 (March 1, 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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8

Chauhan, Vijeyata, and Pankaj Kumar Srivastava. "Computational Techniques Based on Runge-Kutta Method of Various Order and Type for Solving Differential Equations." International Journal of Mathematical, Engineering and Management Sciences 4, no. 2 (April 1, 2019): 375–86. http://dx.doi.org/10.33889/ijmems.2019.4.2-030.

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The Runge-Kutta method is a one step method with multiple stages, the number of stages determine order of method. The method can be applied to work out on differential equation of the type’s explicit, implicit, partial and delay differential equation etc. The present paper describes a review on recent computational techniques for solving differential equations using Runge-Kutta algorithm of various order. This survey includes the summary of the articles of last decade till recent years based on third; fourth; fifth and sixth order Runge-Kutta methods. Along with this a combination of these methods and various other type of Runge-Kutta algorithm based articles are included. Comparisons of methods with own critical comments as remarks have been included.

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Christopher, Dr Esekhaigbe Aigbedion. "Consistency and Convergence Analysis of an 𝐹(𝑥,𝑦) Functionally Derived Explicit Fifth-Stage Fourth-Order Runge-Kutta Method." International Journal of Basic Sciences and Applied Computing 10, no. 4 (December 30, 2022): 10–13. http://dx.doi.org/10.35940/ijbsac.a1145.1210423.

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The purpose of this paper is to analyze the consistency and convergence of an explicit fifth-stage fourth-order Runge-Kutta method derived using 𝒇(𝒙,𝒚) functional derivatives. The analysis revealed that the method is consistent and convergent. The implementation of this method on initial-value problems was done in a previous paper, and it revealed that the method compared favorably well with the existing classical fourth stage fourth order explicit Runge Kutta method.

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10

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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11

Banu, Mst Sharmin. "A Comparative Study on Classical Fourth Order and Butcher Sixth Order Runge-Kutta Methods with Initial and Boundary Value Problems." International Journal of Material and Mathematical Sciences 3, no. 1 (February 28, 2021): 8–21. http://dx.doi.org/10.34104/ijmms.021.08021.

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In this paper, it is discussed about Runge-Kutta fourth-order method and Butcher Sixth order Runge-Kutta method for approximating a numerical solution of higher-order initial value and boundary value ordinary differential equations. The proposed methods are most efficient and extolled practically for solving these problems arising indifferent sector of science and engineering. Also, the shooting method is applied to convert the boundary value problems to initial value problems. Illustrative examples are provided to verify the accuracy of the numerical outcome and compared the approximated result with the exact result. The approximated results are found in good agreement with the result of the exact solution and firstly converge to more accuracy in the solution when step size is very small. Finally, the error with different step sizes is analyzed and compared to these two methods.

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12

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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13

Fajri, Iman Al, Hendra Mesra, and Jeffry Kusuma. "Solving Ordinary Differential Equation Using Parallel Fourth Order Runge-Kutta Method With Three Processors." Jurnal Matematika, Statistika dan Komputasi 17, no. 3 (May 12, 2021): 349–56. http://dx.doi.org/10.20956/j.v17i3.12490.

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This paper presents a derivation of the Runge-Kutta or fourth method with six stages suitable for parallel implementation. Development of a parallel model based on the sparsity structure of the fourth type Runge-Kutta which is divided into three processors. The calculation of the parallel computation model and the sequential model from the accurate side shows that the sequential model is better. However, generally, the parallel method will end the analytic solution by increasing the number of iterations. In terms of execution time, parallel method has advantages over sequential method.

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14

SIMOS, T. E., and JESUS VIGO AGUIAR. "A NEW MODIFIED RUNGE–KUTTA–NYSTRÖM METHOD WITH PHASE-LAG OF ORDER INFINITY FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION AND RELATED PROBLEMS." International Journal of Modern Physics C 11, no. 06 (September 2000): 1195–208. http://dx.doi.org/10.1142/s0129183100001036.

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In this paper, a new approach for developing efficient Runge–Kutta–Nyström methods is introduced. This new approach is based on the requirement of annihilation of the phase-lag (i.e., the phase-lag is of order infinity) and on a modification of Runge–Kutta–Nyström methods. Based on this approach, a new modified Runge–Kutta–Nyström fourth algebraic order method is developed for the numerical solution of Schrödinger equation and related problems. The new method has phase-lag of order infinity and extended interval of periodicity. Numerical illustrations on the radial Schrödinger equation and related problems with oscillating solutions indicate that the new method is more efficient than older ones.

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15

Christopher, Esekhaigbe Aigbedion. "Derivation And Implementation of a Fifth Stage Fourth Order Explicit Runge-Kutta Formula using 𝑓(𝑥,𝑦) Functional Derivatives." Indian Journal of Advanced Mathematics 3, no. 1 (December 30, 2023): 28–33. http://dx.doi.org/10.54105/ijam.a1144.043123.

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This paper is aimed at using 𝒇(𝒙,𝒚) functional derivatives to derive a fifth stage fourth order Explicit Runge-Kutta formula for solving initial value problems in Ordinary Differential Equations. The 𝒇(𝒙,𝒚) functional derivatives from the general Runge-Kutta scheme will be compared with the 𝒇(𝒙,𝒚) functional derivatives from the Taylor series expansion to derive the method. The method will be implemented on some initial value problems, and results compared with results from the classical fourth order method. The results revealed that the method compared favorably well with the existing classical fourth order method.

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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 (May 13, 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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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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Zhang, Lei, Weihua Ou Yang, Xuan Liu, and Haidong Qu. "Fourier Spectral Method for a Class of Nonlinear Schrödinger Models." Advances in Mathematical Physics 2021 (July 1, 2021): 1–11. http://dx.doi.org/10.1155/2021/9934858.

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In this paper, Fourier spectral method combined with modified fourth order exponential time-differencing Runge-Kutta is proposed to solve the nonlinear Schrödinger equation with a source term. The Fourier spectral method is applied to approximate the spatial direction, and fourth order exponential time-differencing Runge-Kutta method is used to discrete temporal direction. The proof of the conservation law of the mass and the energy for the semidiscrete and full-discrete Fourier spectral scheme is given. The error of the semidiscrete Fourier spectral scheme is analyzed in the proper Sobolev space. Finally, several numerical examples are presented to support our analysis.

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19

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 (October 1, 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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Jin Sian, Chai, Yeak Su Hoe, and Ali H. M. Murid. "Some Numerical Methods and Comparisons for Solving Mathematical Model of Surface Decontamination by Disinfectant Solution." MATEMATIKA 34, no. 2 (December 2, 2018): 271–91. http://dx.doi.org/10.11113/matematika.v34.n2.1055.

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A mathematical model is considered to determine the effectiveness of disinfectant solution for surface decontamination. The decontamination process involved the diffusion of bacteria into disinfectant solution and the reaction of the disinfectant killing effect. The mathematical model is a reaction-diffusion type. Finite difference method and method of lines with fourth-order Runge-Kutta method are utilized to solve the model numerically. To obtain stable solutions, von Neumann stability analysis is employed to evaluate the stability of finite difference method. For stiff problem, Dormand-Prince method is applied as the estimated error of fourth-order Runge-Kutta method. MATLAB programming is selected for the computation of numerical solutions. From the results obtained, fourth-order Runge-Kutta method has a larger stability region and better accuracy of solutions compared to finite difference method when solving the disinfectant solution model. Moreover, a numerical simulation is carried out to investigate the effect of different thickness of disinfectant solution on bacteria reduction. Results show that thick disinfectant solution is able to reduce the dimensionless bacteria concentration more effectively

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21

Alhejaili, Weaam, Alvaro H. Salas, Elsayed Tag-Eldin, and Samir A. El-Tantawy. "On Perturbative Methods for Analyzing Third-Order Forced Van-der Pol Oscillators." Symmetry 15, no. 1 (December 29, 2022): 89. http://dx.doi.org/10.3390/sym15010089.

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In this investigation, an (un)forced third-order/jerk Van-der Pol oscillatory equation is solved using two perturbative methods called the Krylov–Bogoliúbov–Mitropólsky method and the multiple scales method. Both the first- and second-order approximations for the unforced and forced jerk Van-der Pol oscillatory equations are derived in detail using the proposed methods. Comparative analysis is performed between the analytical approximations using the proposed methods and the numerical approximations using the fourth-order Runge–Kutta scheme. Additionally, the global maximum error to the analytical approximations compared to the Runge–Kutta numerical approximation is estimated.

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WILLIAMS, P. S., and T. E. SIMOS. "EXPONENTIALLY FITTED RUNGE–KUTTA FOURTH ALGEBRAIC ORDER METHODS FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION AND RELATED PROBLEMS." International Journal of Modern Physics C 11, no. 04 (June 2000): 785–807. http://dx.doi.org/10.1142/s0129183100000687.

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Fourth order exponential and trigonometric fitted Runge–Kutta methods are developed in this paper. They are applied to problems involving the Schrödinger equation and to other related problems. Numerical results show the superiority of these methods over conventional fourth order Runge–Kutta methods. Based on the methods developed in this paper, a variable-step algorithm is proposed. Numerical experiments show the efficiency of the new algorithm.

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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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Morlando, Fabrizio. "Approximate Analytical Solution of a Third-Order IVP arising in Thin Film Flows driven by Surface Tension." Boletim da Sociedade Paranaense de Matemática 35, no. 3 (October 25, 2017): 117–29. http://dx.doi.org/10.5269/bspm.v35i3.28349.

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In this paper, we present a way of applying the so-called He's variational iteration method (VIM) to numerically solve the non linear autonomous third-order ordinary dierential equation (ODE) y''' = y^-2 obtained by considering a traveling wave solution admitted by a lubrication equation modeling a two-dimensional spreading of a thin viscous lm on a inclined slope. Approximate analytical solution is derived and compared to the results obtained from the Adomian decomposition method (ADM) proposed in [20], to the exact analytical solution[7,8], to a fth order Runge-Kutta method (DOPRI), a fourth order Runge-Kutta method (RK4), a three-stage fth order Runge-Kutta method (RKD5) developed in [18]. A very good agreement and accuracy is observed. Comparisons are obtained using symbolic capabilities of Maple 18.0 package.

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Musa, H., Ibrahim Saidu, and M. Y. Waziri. "A Simplified Derivation and Analysis of Fourth Order Runge Kutta Method." International Journal of Computer Applications 9, no. 8 (November 10, 2010): 51–55. http://dx.doi.org/10.5120/1402-1891.

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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 (November 20, 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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Yaakub, A. R., and D. J. Evans. "A fourth order Runge–Kutta RK(4,4) method with error control." International Journal of Computer Mathematics 71, no. 3 (January 1999): 383–411. http://dx.doi.org/10.1080/00207169908804817.

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Bylina, Beata, and Joanna Potiopa. "Explicit Fourth-Order Runge–Kutta Method on Intel Xeon Phi Coprocessor." International Journal of Parallel Programming 45, no. 5 (September 29, 2016): 1073–90. http://dx.doi.org/10.1007/s10766-016-0458-x.

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Fok, Pak-Wing. "A Linearly Fourth Order Multirate Runge–Kutta Method with Error Control." Journal of Scientific Computing 66, no. 1 (March 28, 2015): 177–95. http://dx.doi.org/10.1007/s10915-015-0017-4.

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Ramli, A., R. R. Ahmad, and U. K. S. Din. "MODIFIED FOURTH ORDER RUNGE-KUTTA METHOD FOR SOLVING FUZZY DIFFERENTIAL EQUATIONS." Far East Journal of Mathematical Sciences (FJMS) 101, no. 10 (May 19, 2017): 2299–315. http://dx.doi.org/10.17654/ms101102299.

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Kalogiratou, Z., Th Monovasilis, and T. E. Simos. "A fourth order modified trigonometrically fitted symplectic Runge–Kutta–Nyström method." Computer Physics Communications 185, no. 12 (December 2014): 3151–55. http://dx.doi.org/10.1016/j.cpc.2014.08.013.

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32

Walters, Stephen J., Ross J. Turner, and Lawrence K. Forbes. "A comparison of explicit Runge-Kutta methods." ANZIAM Journal 64 (March 19, 2023): 227–49. http://dx.doi.org/10.21914/anziamj.v64.17438.

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Recent higher-order explicit Runge–Kutta methods are compared with the classic fourth-order (RK4) method in long-term integration of both energy-conserving and lossy systems. By comparing quantity of function evaluations against accuracy for systems with and without known solutions, optimal methods are proposed. For a conservative system, we consider positional accuracy for Newtonian systems of two or three bodies and total angular momentum for a simplified Solar System model, over moderate astronomical timescales (tens of millions of years). For a nonconservative system, we investigate a relativistic two-body problem with gravitational wave emission. We find that methods of tenth and twelfth order consistently outperform lower-order methods for the systems considered here. doi: 10.1017/S1446181122000141

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Senu, Norazak, Nur Amirah Ahmad, Zarina Bibi Ibrahim, and Mohamed Othman. "Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS." Sains Malaysiana 50, no. 6 (June 30, 2021): 1799–814. http://dx.doi.org/10.17576/jsm-2021-5006-25.

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A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

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34

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 (December 29, 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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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 (November 2, 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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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 (April 17, 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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Sharmila, R. Gethsi, and E. C. Henry Amirtharaj. "Numerical Solution of Fuzzy Initial Value Problems by Fourth Order Runge-Kutta Method Based on Contraharmonic Mean." Indian Journal of Applied Research 3, no. 4 (October 1, 2011): 59–63. http://dx.doi.org/10.15373/2249555x/apr2013/111.

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Akogwu, Blessing. "THE SOLUTION OF A MATHEMATICAL MODEL FOR COVID-19 TRANSMISSION AND VACCINATION IN NIGERIA BY USING A DIFFERENTIAL TRANSFORMATION METHOD." FUDMA JOURNAL OF SCIENCES 6, no. 5 (November 2, 2022): 50–56. http://dx.doi.org/10.33003/fjs-2022-0605-1089.

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In this work, Differential Transform Method (DTM) was employed to obtain the series solution of the SIRV COVID-19 model in Nigeria. The validity of the DTM in solving the model was validated by Maple 21’s Classical fourth-order Runge-Kutta method. The comparison between DTM and Runge- Kutta (RK4) solutions was performed and there was a good correlation between the results obtained by the two methods. The result validates the accuracy and efficiency of the DTM to solve the model

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SIMOS, T. E., and P. S. WILLIAMS. "SOME MODIFIED RUNGE–KUTTA METHODS FOR THE NUMERICAL SOLUTION OF SOME SPECIFIC SCHRÖDINGER EQUATIONS AND RELATED PROBLEMS." International Journal of Modern Physics A 11, no. 26 (October 20, 1996): 4731–44. http://dx.doi.org/10.1142/s0217751x96002169.

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Some new modified Runge–Kutta methods with minimal phase lag are developed for the numerical solution of the eigenvalue Schrödinger equation and related problems with oscillating solutions. These methods are based on the very well-known Runge–Kutta method of order 4. For the numerical solution of the eigenvalue Schrödinger equation, we investigate two cases: (i) the specific case in which the potential V(x) is an even function with respect to x; it is assumed, also, that the wave functions tend to zero for x → ±∞; (ii) the general case for the well-known cases of the Morse potential and Woods–Saxon or optical potential. Also, we have applied the new methods to some well-known problems with oscillatory solutions. Numerical and theoretical results show that this new approach is more efficient than the well-known classical fourth order Runge–Kutta method and the Numerov method.

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Izzah, Nor Atirah, Yeak Su Hoe, and Normah Maan. "Fast and Robust Parameter Estimation in the Application of Fuzzy Logistic Equations in Population Growth." MATEMATIKA 35, no. 2 (July 31, 2019): 249–59. http://dx.doi.org/10.11113/matematika.v35.n2.1164.

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In this paper, extended Runge-Kutta fourth order method for directly solving the fuzzy logistic problem is presented. The extended Runge-Kutta method has lower number of function evaluations, compared with the classical Runge-Kutta method. The numerical robustness of the method in parameter estimation is enhanced via error minimization in predicting growth rate and carrying capacity. The results of fuzzy logistic model with the estimated parameters have been compared with population growth data in Malaysia, which indicate that this method is more accurate that the data population. Numerical example is given to illustrate the efficiency of the proposed model. It is concluded that robust parameter estimation technique is efficient in modelling population growth.

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MUKHERJEE, SUPRIYA, DEBKALPA GOSWAMI, and BANAMALI ROY. "SOLUTION OF HIGHER-ORDER ABEL EQUATIONS BY DIFFERENTIAL TRANSFORM METHOD." International Journal of Modern Physics C 23, no. 09 (September 2012): 1250056. http://dx.doi.org/10.1142/s0129183112500568.

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In this paper, the second-, third- and fourth-order Abel equations are solved. The differential transform method (DTM) is used to compute approximate solutions of the nonlinear ordinary Abel differential equations. The results are compared with the results obtained by the classical Runge–Kutta (RK4) method. Figures are presented to show the reliability and simplicity of the method.

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Side, Syafruddin, Ahmad Zaki, and Miswar. "Numerical Solution of the Mathematical Model of DHF Spread using the Runge-Kutta Fourth Order Method." ARRUS Journal of Mathematics and Applied Science 2, no. 2 (April 5, 2022): 92–100. http://dx.doi.org/10.35877/mathscience745.

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This research was conducted to find a numerical solution to the mathematical model of DHF in Makassar using the Runge-Kutta fourth order method. The mathematical model of DHF is in the form of a system of differential equations that includes variables S (Susceptible), E (Exposed), I (Infected), and R (Recovery) simplified into classes of vulnerable (S), exposed (E), infected (I) and cured (R) as initial value. Parameters value that is solved numerically using the Runge-Kutta fourth order method with time intervals h = 0.01 months using data from South Sulawesi Provincial Health Service in 2017. Based on the initial value of each class, namely: obtained (Sh1) =10910.4, (E) = 0, (Ih1) = 177.9 , (Sv1) = 5018685.6, (Iv1) = 135.4, and R = -981612.3. The initial values and parameter values are substituted into numerical solutions to the model simulated using maple as a tool.

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Skvortsov, L. M. "Singly implicit diagonally extended Runge-Kutta methods of fourth order." Computational Mathematics and Mathematical Physics 54, no. 5 (May 2014): 775–84. http://dx.doi.org/10.1134/s0965542514050133.

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44

Tang, Jianhua, Linfang Qian, and Qiang Yin. "Numerical integration of rotation with geometry propagators." International Journal of Modeling, Simulation, and Scientific Computing 07, no. 03 (August 23, 2016): 1650017. http://dx.doi.org/10.1142/s1793962316500173.

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Commutator free method is an effective method for solving rotating integration. Numerical examples show that the use of the proposed combining method can achieve the same order accuracy with less computation than other geometry integration method. However, it is difficult to be directly applied to mechanic dynamics solutions. In this paper, commutator free method which is often applied to rotation integration and classical Runge–Kutta (RK) method which is usually operated in Linear space are combined to solve the multi-body dynamic equations. The explicit Runge–Kutta coefficients are reconstructed to meet different order accuracy integration methods. The reconstruction method is discussed and coefficients are given. With this method, the dynamic equations can be solved accurately and economically without much modification on the classical numerical integration. Moreover, CG method and CF method can also be combined with adaptive RK method without many changes. Finally, the results of the examples show that with less computation, fourth-order combining method is as accurate as fourth-order Crouch–Grossman algorithm.

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45

Nurofi’atin, Umi, and Agus Maman Abadi. "Model Analysis of Motorcycle Suspension System Using the Fourth Order of Runge-Kutta Method." JURNAL EKSAKTA 18, no. 2 (September 17, 2018): 106–20. http://dx.doi.org/10.20885/eksakta.vol18.iss2.art3.

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The suspension system is part of motorcycle that serves to absorb vibration and shocks of the road surface so as to improve the safety and comfort while driving. Motorcycle typically use double shockbreaker system which analogous to a two-spring system arranged in parallel. The aim of this researh is to analyze the model of the model of double shockbreaker motorcycle suspension system that working without outside force using passive suspension system. The data used are from damper tester experiment, then model analyzed using analytical method and the fourth order of numerical Runge-Kutta method. This research use shockbreaker observation datas that is the measurment data of spring constant and damping constant by performing damper tester using 4 different loads. The process model analysis using Matlab R2013a. Input variables are spring constant, damping constant, and the mass of the load. Methods of analysis using analytical method and the fourth order of Runge-Kutta method. While the resulting outputs are 2 spring constants, change the length of the spring, damping ratio, the optimal damping of the suspension, and the spring deflection chart against time. This model motorcycle suspension system uses solution of differential equations for the under damped suspension condition, that is the suspension system will be insulated a few moments before reaching the equilibrium position. Therefore, the resulting damping rate of the motorcycle is not optimal yet. This study found the optimal damping for each model of the suspension system. The level of accuracy of the fourth order of runge-kutta method for model analysis of the suspension system is quite high with error <0.1 and the timing of analysis is faster than the analytic method. Future research may use other methods or other input variables for more accurate analysis results.

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Sukron, M., U. Habibah, and N. Hidayat. "Numerical solution of Saint-Venant equation using Runge-Kutta fourth-order method." Journal of Physics: Conference Series 1872, no. 1 (May 1, 2021): 012036. http://dx.doi.org/10.1088/1742-6596/1872/1/012036.

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Evans, D. J., and N. Yaacob. "A fourth order runge-kutta method based on the heronian mean formula." International Journal of Computer Mathematics 58, no. 1-2 (January 1995): 103–15. http://dx.doi.org/10.1080/00207169508804437.

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Ramos, Higinio, and Jesús Vigo-Aguiar. "A fourth-order Runge–Kutta method based on BDF-type Chebyshev approximations." Journal of Computational and Applied Mathematics 204, no. 1 (July 2007): 124–36. http://dx.doi.org/10.1016/j.cam.2006.04.033.

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Mulyadi, Lutfi, and Zaki Su’ud. "Fast Nuclear Reactor Fuel Depletion Analysis Using Fourth Order Runge-Kutta Method." Journal of Physics: Conference Series 1493 (March 2020): 012014. http://dx.doi.org/10.1088/1742-6596/1493/1/012014.

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50

Demendy, Zoltan. "Fourth order Runge-Kutta method in 2d for linear boundary value problems." International Journal for Numerical Methods in Engineering 32, no. 6 (October 25, 1991): 1229–45. http://dx.doi.org/10.1002/nme.1620320605.

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