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Journal articles on the topic 'Runge-Kutta metody'

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Published: 4 June 2021

Last updated: 17 February 2022

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1

Jackiewicz, Zdzislaw, Rosemary Anne Renaut, and Marino Zennaro. "Explicit two-step Runge-Kutta methods." Applications of Mathematics 40, no. 6 (1995): 433–56. http://dx.doi.org/10.21136/am.1995.134306.

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2

Hidayat, Taufik, Jenizon ., and Budi Rudianto. "APLIKASI PRINSIP MAKSIMUM PONTRYAGIN DAN METODE RUNGE-KUTTA DALAM MASALAH KONTROL OPTIMAL." Jurnal Matematika UNAND 7, no. 2 (May 1, 2018): 212. http://dx.doi.org/10.25077/jmu.7.2.212-220.2018.

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Abstrak. Tulisan ini membahas tentang penggunaan prinsip maksimum pontryagin danmetode Runge-Kutta dalam masalah kontrol optimal. Prinsip maksimum pontryagin di-gunakan untuk menentukan solusi analitik, dan metode Runge-Kutta untuk menentukansolusi numerik. Hasil dari metode Runge-Kutta kemudian dibandingkan dengan solusianalitiknya. Dari hasil perbandingan dapat disimpulkan metode Runge-Kutta mem-berikan galat yang sangatlah kecil dan dapat diabaikan.Kata Kunci: Masalah Kontrol Optimal, Prinsip Maksimum Pontryagin, Runge-Kutta

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3

Segarra, Jaime. "MÉTODOS NUMÉRICOS RUNGE-KUTTA Y ADAMS BASHFORTH-MOULTON EN MATHEMATICA." Revista Ingeniería, Matemáticas y Ciencias de la Información 7, no. 14 (July 15, 2020): 13–32. http://dx.doi.org/10.21017/rimci.2020.v7.n14.a81.

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n este estudio, el objetivo principal es realizar el análisis de los métodos numéricos Runge-Kutta y Adams Bashforth-Moulton. Para cumplir con el objetivo se utilizó el sistema de ecuaciones diferenciales del modelo Lotka-Volterra y se usó el software matemático Wolfram Mathematica. En los resultados se realiza la comparación de los métodos RK4, AB4 y AM4 con el comando NDSolve utilizando el modelo Lotka-Volterra. Los resultados obtenidos en los diagramas de fase y la tabla de puntos de la iteración indicaron que el método RK4 tiene mayor precisión que los métodos AB4 y AM4.

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4

Acu, Yulia, Boni Pahlanop Lapanporo, and Arie Antasari Kushadiwijayanto. "Model Sederhana Gerak Osilator dengan Massa Berubah Terhadap Waktu Menggunakan Metode Runge Kutta." POSITRON 7, no. 2 (January 6, 2018): 42. http://dx.doi.org/10.26418/positron.v7i2.23276.

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Persamaan gerak osilasi pegas dengan massa berubah terhadap waktu merupakan persamaan differensial non linear yang sulit diselesaikan secara analitik. Pada penelitian ini persamaan gerak osilasi tersebut diselesaikan menggunakan metode Runge Kutta orde empat dan Runge Kutta Fehlberg tingkat lima. Metode Runge Kutta menawarkan penyelesaian persamaan diferensial dengan pertumbuhan truncation error yang jauh lebih kecil. Hasilnya menunjukan gerak osilator dengan massa yang terus berkurang terhadap waktu memiliki sifat seperti osilator teredam dan menjadi gerak harmonik sederhana saat massa osilator tetap. Metode Runge Kutta orde empat dan Runge Kutta Fehlberg mampu menggambarkan keadaan sistem osilator dengan massa berubah terhadap waktu.

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5

Rahmatullah, Sigid, Yudha Arman, and Apriansyah Apriansyah. "Simulasi Gerak Osilasi Model Pegas Bergandeng Menggunakan Metode Runge-Kutta." PRISMA FISIKA 8, no. 3 (December 15, 2020): 180. http://dx.doi.org/10.26418/pf.v8i3.43681.

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Persamaan gerak sistem pegas bergandeng Fay dan Graham (2003) telah diselesaikan secara numerik untuk kemudian dibandingkan dengan hasil perhitungan analitik. Nilai kesebandingan didasarkan pada Symmetric Mean Absolute Percentage Error (SMAPE). Metode numerik utama yang digunakan untuk menyelesaikan persamaan gerak model pegas bergandeng adalah Runge-Kutta Orde Empat dan Runge-Kutta 45 Fehlberg sedangkan metode Leapfrog dan Euler digunakan sebagai metode tambahan uji. Ukuran langkah h yang digunakan adalah 0,05 s. Berdasarkan hasil perhitungan dan nilai SMAPE yang diperoleh, Runge-Kutta 45 Fehlberg menjadi metode numerik dengan tingkat ketelitian yang paling baik diantara berbagai metode numerik yang digunakan pada konfigurasi sistem pegas hasil modifikasi model Fay dan Graham (2003) dengan variasi massa m, arah simpangan x, dan besar konstanta pegas k.Kata Kunci : Pegas, Runge Kutta 45 Fehlberg

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6

Hurit, Roberta Uron, and Sudi Mungkasi. "The Euler, Heun, and Fourth Order Runge-Kutta Solutions to SEIR Model for the Spread of Meningitis Disease." Mathline : Jurnal Matematika dan Pendidikan Matematika 6, no. 2 (August 2, 2021): 140–53. http://dx.doi.org/10.31943/mathline.v6i2.176.

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Penelitian ini bertujuan untuk menyelesaikan model SEIR penyebaran penyakit meningitis menggunakan metode Euler, metode Heun dan metode Runge-Kutta orde empat, dimana model persamaannya berupa persamaan diferensial nonlinear. Metode penelitian yang digunakan adalah pemrograman komputer dan simulasi. Dari simulasi, ketiga metode tersebut menghasilkan penyelesaian dengan perilaku yang mirip, yaitu semua gambar grafik pada setiap simulasi mempunyai bentuk pola yang sama. Hal ini memberikan kepercayaan pada kebenaran hasil simulasi dalam makalah ini. Secara teori, metode Runge-Kutta orde empat memiliki ketelitian yang lebih tinggi dibandingkan dengan metode Euler dan metode Heun. Kata Kunci: penyebaran penyakit meningitis, model SEIR, metode Euler, metode Heun, metode Runge-Kutta orde empat.

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7

Zheng, Zheming, and Linda Petzold. "Runge–Kutta–Chebyshev projection method." Journal of Computational Physics 219, no. 2 (December 2006): 976–91. http://dx.doi.org/10.1016/j.jcp.2006.07.005.

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8

Tan, Jia Bo. "Symplectic Partitioned Runge-Kutta and Symplectic Runge-Kutta Methods Generated by 2-Stage RadauIA Method." Applied Mechanics and Materials 444-445 (October 2013): 633–36. http://dx.doi.org/10.4028/www.scientific.net/amm.444-445.633.

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To preserve the symplecticity property, it is natural to require numerical integration of Hamiltonian systems to be symplectic. As a famous numerical integration, it is known that the 2-stage RadauIA method is not symplectic. With the help of symplectic conditions of Runge-Kutta method and partitioned Runge-Kutta method, a symplectic partitioned Runge-Kutta method and a symplectic Runge-Kutta method are constructed on the basis of 2-stage RadauIA method in this paper.

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9

Muhammad, Raihanatu. "THE ORDER AND ERROR CONSTANT OF A RUNGE-KUTTA TYPE METHOD FOR THE NUMERICAL SOLUTION OF INITIAL VALUE PROBLEM." FUDMA JOURNAL OF SCIENCES 4, no. 2 (October 13, 2020): 743–48. http://dx.doi.org/10.33003/fjs-2020-0402-256.

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Implicit Runge- Kutta methods are used for solving stiff problems which mostly arise in real life situations. Analysis of the order, error constant, consistency and convergence will help in determining an effective Runge- Kutta Method (RKM) to use. Due to the loss of linearity in Runge –Kutta Methods and the fact that the general Runge –Kutta Method makes no mention of the differential equation makes it impossible to define the order of the method independently of the differential equation. In this paper, we examine in simpler details how to obtain the order, error constant, consistency and convergence of a Runge -Kutta Type method (RKTM) when the step number .

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10

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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11

Lobão, Diomar Cesar. "Low storage explicit Runge-Kutta method." Semina: Ciências Exatas e Tecnológicas 40, no. 2 (December 18, 2019): 123. http://dx.doi.org/10.5433/1679-0375.2019v40n2p123.

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Neste artigo estamos tratando dos métodos explícitos Runge Kutta (LSERK) de alta ordem e baixo armazenamento, que são usados principalmente para a discretização temporal e são estáveis independentemente de sua precisão. O principal objetivo deste trabalho é comparar o RK tradicional com diferentes formas de métodos LSERK. Os experimentos numéricos indicam que tais métodos são altamente precisos e eficazes para propósitos numéricos. Também é mostrado o tempo de CPU e suas implicações na solução. O método é bem adequado para obter uma solução precisa de alta ordem para o problema escalar de segunda ordem do problema de valor inicial (IVP), como é discutido no presente artigo.

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12

Tiwari, Shruti, and Ram K. Pandey. "Exponentially-fitted pseudo Runge-Kutta method." International Journal of Computing Science and Mathematics 12, no. 2 (2020): 105. http://dx.doi.org/10.1504/ijcsm.2020.10033205.

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13

Tiwari, Shruti, and Ram K. Pandey. "Exponentially-fitted pseudo Runge-Kutta method." International Journal of Computing Science and Mathematics 12, no. 2 (2020): 105. http://dx.doi.org/10.1504/ijcsm.2020.111118.

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14

Coulibaly, Ibrahim, and Christian Lécot. "A quasi-randomized Runge-Kutta method." Mathematics of Computation 68, no. 226 (April 1, 1999): 651–60. http://dx.doi.org/10.1090/s0025-5718-99-01056-x.

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15

Evans, D. J., and B. B. Sanugi. "A parallel Runge-Kutta integration method." Parallel Computing 11, no. 2 (August 1989): 245–51. http://dx.doi.org/10.1016/0167-8191(89)90032-x.

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16

Hasan, M. Kamrul, M. Suzan Ahamed, M. S. Alam, and M. Bellal Hossain. "An Implicit Method for Numerical Solution of Singular and Stiff Initial Value Problems." Journal of Computational Engineering 2013 (September 26, 2013): 1–5. http://dx.doi.org/10.1155/2013/720812.

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An implicit method has been presented for solving singular initial value problems. The method is simple and gives more accurate solution than the implicit Euler method as well as the second order implicit Runge-Kutta (RK2) (i.e., implicit midpoint rule) method for some particular singular problems. Diagonally implicit Runge-Kutta (DIRK) method is suitable for solving stiff problems. But, the derivation as well as utilization of this method is laborious. Sometimes it gives almost similar solution to the two-stage third order diagonally implicit Runge-Kutta (DIRK3) method and the five-stage fifth order diagonally implicit Runge-Kutta (DIRK5) method. The advantage of the present method is that it is used with less computational effort.

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17

Sun, Jian-Qiang, Rong-Fang Huang, Xiao-Yan Gu, and Ling Yu. "Lie Group Method of the Diffusion Equations." Advances in Mathematical Physics 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/641918.

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The diffusion equation is discretized in spacial direction and transformed into the ordinary differential equations. The ordinary differential equations are solved by Lie group method and the explicit Runge-Kutta method. Numerical results showed that Lie group method is more stable than the corresponding explicit Runge-Kutta method.

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18

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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19

Side, Syafruddin, Maya Sari Wahyuni, and Arifuddin R. "Solusi Numerik Model Verhulst pada Estimasi Pertumbuhan Hasil Panen Padi dengan Metode Adam Bashforth-Moulton (ABM)." Journal of Mathematics, Computations, and Statistics 2, no. 1 (May 12, 2020): 91. http://dx.doi.org/10.35580/jmathcos.v2i1.12463.

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Penelitian ini menerapkan metode Adam Bashforth-Moulton untuk menentukan solusi model Verhulst. Bentuk solusi yang diperoleh adalah estimasi hasil panen padi di Kabupaten Gowa dengan menggunakan persamaan berikut . Persamaan model Verhulst terlebih dahulu diselesaikan dengan metode Runge-Kutta orde-4 untuk mendapatkan solusi awal ; ; dan . Selanjutnya nilai awal disubstitusi pada persamaan Adam-Bashforth orde-4 untuk mendapatkan nilai prediksi, kemudian nilai prediksi yang diperoleh diperbaiki menggunakan persamaan korektor Adam Moulton orde-4. Pada iterasi ke-14 yaitu saat menunjukkan tahun diperoleh nilai prediktor dan nilai korektor sehingga estimasi hasil panen padi di Kabupaten Gowa pada tahun 2021 dengan menggunakan metode Adam Bashforth-Moulton saat adalah ton.Kata Kunci: Model Verhulst, Metode Runge-Kutta, Metode Adam Bashforth-Moulton This research applied Adam Bashforth-Moulton Method to determine the solution of Verhust Model. The form of the solution obtained is estimatation of rice harvest in Gowa Regency by using the following equation . Verhulst model equation firstly solved by using 4th order of Runge-Kutta method to get initial solutions of ; ; and . Furthermore, the initial values subtituted on the 4th order of Adam-Bashforth equation to get the prediction value, then the prediction value obtained was corrected using the corrector equation of 4th order of Adam Moulton. On the 14th iteration that is when shows the year of 2021 retrieved the predictor value of and corrector value of so estimation of rice harvets in Gowa Regency in 2021 by using Adam Bashforth-Moulton method when is ton.Keywords: Verhulst Model, Runge-Kutta Method, Adam Bashforth-Moulton

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20

Chowdhury, Abhinandan, Sammie Clayton, and Mulatu Lemma. "Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method." JOURNAL OF ADVANCES IN MATHEMATICS 17 (September 16, 2019): 147–54. http://dx.doi.org/10.24297/jam.v17i0.8408.

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We present a study on numerical solutions of nonlinear ordinary differential equations by applying Runge-Kutta-Fehlberg (RKF) method, a well-known adaptive Runge-kutta method. The adaptive Runge-kutta methods use embedded integration formulas which appear in pairs. Typically adaptive methods monitor the truncation error at each integration step and automatically adjust the step size to keep the error within prescribed limit. Numerical solutions to different nonlinear initial value problems (IVPs) attained by RKF method are compared with corresponding classical Runge-Kutta (RK4) approximations in order to investigate the computational superiority of the former. The resulting gain in efficiency is compatible with the theoretical prediction. Moreover, with the aid of a suitable time-stepping scheme, we show that the RKF method invariably requires less number of steps to arrive at the right endpoint of the finite interval where the IVP is being considered.

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21

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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22

M. Ismail, Mohammed. "Goeken-Johnson Sixth-Order Runge-Kutta Method." JOURNAL OF EDUCATION AND SCIENCE 24, no. 1 (March 1, 2011): 119–28. http://dx.doi.org/10.33899/edusj.2011.51502.

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23

Virk, G. S. "Runge Kutta method for delay-differential systems." IEE Proceedings D Control Theory and Applications 132, no. 3 (1985): 119. http://dx.doi.org/10.1049/ip-d.1985.0021.

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24

Wazwaz, Abdul-Majid. "A modified third order Runge-Kutta method." Applied Mathematics Letters 3, no. 3 (1990): 123–25. http://dx.doi.org/10.1016/0893-9659(90)90154-4.

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25

Khalsaraei, M. Mehdizadeh. "Positivity of an explicit Runge–Kutta method." Ain Shams Engineering Journal 6, no. 4 (December 2015): 1217–23. http://dx.doi.org/10.1016/j.asej.2015.05.018.

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26

Ixaru, L. Gr. "Runge–Kutta method with equation dependent coefficients." Computer Physics Communications 183, no. 1 (January 2012): 63–69. http://dx.doi.org/10.1016/j.cpc.2011.08.017.

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27

Ariefatosa, Firdaus, Budi Santoso, Muzilman Muslim, and Ari Mutanto. "Penyelesaian Numerik Hamburan Kuantum Potensial Sentral Dengan Metode Runge-Kutta." Jurnal Ilmiah Giga 17, no. 1 (March 20, 2019): 38. http://dx.doi.org/10.47313/jig.v17i1.537.

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Telah dilakukan perhitungan tampang lintang total hamburan dengan metode gelombang parsial. Dalam metode ini, besaran yang menjadi kunci adalah geser fasa δl diperoleh dengan menyelesaikan persamaan diferensial bagian radial hamburan kuantum dengan dan tanpa potensial penghambur. Perhitungan Numerik yang dipilih adalah metode Runge-Kutta karena kemudahan aplikasi dan akurasi tinggi. Potensial hamburan yang digunakan adalah potensial Coulomb tertirai (Screened-Coulomb) yang telah diketahui parameter-parameter potensialnya sebagaimana telah disajikan oleh Salvat (Salvat et al, 1987) melalui metode variasi medan konsisten. Hamburan yang dimaksud adalah hamburan elektron-atom.

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28

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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29

Nur, Wahyudin, and Nurul Mukhlisah Abdal. "Solusi Numerik Model Umum Epidemik Susceptible, Infected, Recovered (SIR) dengan Menggunakan Metode Modified Milne-Simpson." SAINTIFIK 2, no. 2 (July 2, 2016): 142–46. http://dx.doi.org/10.31605/saintifik.v2i2.159.

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Model Epidemik Susceptible, Infected, Recovered (SIR) merupakan salah satu metode yang paling banyak digunakan untuk memodelkan penyebaran penyakit. Model ini biasa digunakan untuk simulasi dan ptediksi jumlah kasus penyakit tertentu. Dalam artikel ini penulis melakukan simulasi dan mencari solusi numerik model umum epidemik SIR dengan menggunakan MMetode Modified Milne Simpson yang dipadukan dengan metode Runge Kutta Orde 4. Metode ini merupakan salah satu metode prediktor korektor yang biasa digunakan untuk mencari solusi numerik persamaan diferensial. Dengan menggunakan parameter miu=0,1;lamda=0,0098; gamma=0,5 diperoleh r0=0,016333<1. Kurva kelas Infected menuju nol dan setimbang dititik nol. Hal ini menandakan, dengan pemilihan parameter seperti itu, kelas Infected akan menghilang dari populasi. Berdasarkan hasil simulasi, dapat disipulkan bahwa metode Milne Simpson layak digunakan untuk menentukan solusi numerik model umum epidemik SIR.Kata kunci: Model SIR, Modified Milnw Simpson, Runge Kutta Orde 4

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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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31

Abassy, Tamer. "Piecewise Analytic Method VS Runge-Kutta Method (Comparative Study)." International Journal of Applied Mathematical Research 9, no. 2 (October 23, 2020): 41. http://dx.doi.org/10.14419/ijamr.v9i2.31118.

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Even though Runge-Kutta (RK) method is the most used by scientists and engineers, it is not the most powerful method. In this paper, a comparative study between Piecewise Analytic Method (PAM) and RK methods is achieved. The result of comparative study shows that PAM is more powerful and gives results better than RK Methods. PAM can be considered as a new step in the evolution of solving nonlinear differential equations.

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32

Michael, Akpini K. A., Assui K. Richard, Yoro Gozo, and Bailly Bale. "Optimal Method of Runge-Kutta of Order 5." Journal of Mathematics Research 11, no. 1 (January 24, 2019): 93. http://dx.doi.org/10.5539/jmr.v11n1p93.

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The Runge-Kutta method of order 5 with 6 stages requires finding a matrix A, whose coefficients must satisfy a system of nonlinear polynomial equations. Butcher found a 5-parameter family of solutions, which displays different characteristics depending on whether b2= 0 or b2, 0. This paper presents an optimal method in the case b2= 0, which is significantly better than several popular methods of order 4.

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Hippolyte, Séka, and Assui Kouassi Richard. "A New Eighth Order Runge-Kutta Family Method." Journal of Mathematics Research 11, no. 2 (March 25, 2019): 190. http://dx.doi.org/10.5539/jmr.v11n2p190.

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In this article, a new family of Runge-Kutta methods of 8^th order for solving ordinary differential equations is discovered and depends on the parameters b_8 and a_10;5. For b8 = 49/180 and a10;5 = 1/9, we find the Cooper-Verner method [1]. We show that the stability region depends only on coefficient a_10;5. We compare the stability regions according to the values of a_10;5 with respect to the stability region of Cooper-Verner.

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34

Usha, B., AL Nachammai, and M. Dhavamani. "Solving Fuzzy Differential Equations by Runge-Kutta Method." Asian Journal of Research in Social Sciences and Humanities 6, no. 9 (2016): 147. http://dx.doi.org/10.5958/2249-7315.2016.00784.x.

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35

Janezic, Dusanka, and Bojan Orel. "Implicit Runge-Kutta method for molecular dynamics integration." Journal of Chemical Information and Modeling 33, no. 2 (March 1, 1993): 252–57. http://dx.doi.org/10.1021/ci00012a011.

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36

Akbarzadeh Ghanaie, Z., and M. Mohseni Moghadam. "Solving Fuzzy Differential Equations By Runge-kutta Method." Journal of Mathematics and Computer Science 02, no. 02 (February 28, 2011): 208–21. http://dx.doi.org/10.22436/jmcs.002.02.01.

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IMAI, Yohsuke, Takayuki Aoki, and Tetsuya Kobara. "Implicit IDO scheme by using Runge-Kutta method." Proceedings of The Computational Mechanics Conference 2003.16 (2003): 151–52. http://dx.doi.org/10.1299/jsmecmd.2003.16.151.

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Wusu, Ashiribo Senapon, Moses Adebowale Akanbi, and Solomon Adebola Okunuga. "A Three-Stage Multiderivative Explicit Runge-Kutta Method." American Journal of Computational Mathematics 03, no. 02 (2013): 121–26. http://dx.doi.org/10.4236/ajcm.2013.32020.

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Evans, D. J., and A. R. Yaakub. "A new Runge Kutta RK(4, 4) method." International Journal of Computer Mathematics 58, no. 3-4 (January 1995): 169–87. http://dx.doi.org/10.1080/00207169508804442.

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Riza, Mustafa, and Hatice Aktöre. "The Runge–Kutta method in geometric multiplicative calculus." LMS Journal of Computation and Mathematics 18, no. 1 (2015): 539–54. http://dx.doi.org/10.1112/s1461157015000145.

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This paper illuminates the derivation, applicability, and efficiency of the multiplicative Runge–Kutta method, derived in the framework of geometric multiplicative calculus. The removal of the restrictions of geometric multiplicative calculus on positive-valued functions of real variables and the fact that the multiplicative derivative does not exist at the roots of the function are presented explicitly to ensure that the proposed method is universally applicable. The error and stability analyses are also carried out explicitly in the framework of geometric multiplicative calculus. The method presented is applied to various problems and the results are compared to those obtained from the ordinary Runge–Kutta method. Moreover, for one example, a comparison of the computation time against relative error is worked out to illustrate the general advantage of the proposed method.

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Qiu., Jianxian, and Chi-Wang Shu. "Runge--Kutta Discontinuous Galerkin Method Using WENO Limiters." SIAM Journal on Scientific Computing 26, no. 3 (January 2005): 907–29. http://dx.doi.org/10.1137/s1064827503425298.

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Verwer, J. G., W. H. Hundsdorfer, and B. P. Sommeijer. "Convergence properties of the Runge-Kutta-Chebyshev method." Numerische Mathematik 57, no. 1 (December 1990): 157–78. http://dx.doi.org/10.1007/bf01386405.

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Srivastava, A., and S. L. Paveri-Fontana. "Numerical experiments on the rational Runge-Kutta method." Computers & Mathematics with Applications 12, no. 12 (December 1986): 1161–70. http://dx.doi.org/10.1016/0898-1221(86)90001-5.

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Sugiura, Hiroshi, and Tatsuo Torii. "A method for constructing generalized Runge-Kutta methods." Journal of Computational and Applied Mathematics 38, no. 1-3 (December 1991): 399–410. http://dx.doi.org/10.1016/0377-0427(91)90185-m.

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Öncü, Selim, Kenan Ünal, and Uğur Tunçer. "D Sınıfı Eviricinin Runge-Kutta Metodu ile Analizinin Yapılması." Academic Perspective Procedia 3, no. 1 (October 25, 2020): 337–45. http://dx.doi.org/10.33793/acperpro.03.01.66.

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D sınıfı eviriciler geniş kontrol aralığına sahip olmaları, az sayıda g&uuml;&ccedil; anahtarı ve anahtar s&uuml;r&uuml;c&uuml; devre elemanları gerektirmeleri ve y&uuml;ksek g&uuml;&ccedil;leri kontrol edebilmeleri sebebiyle g&uuml;n&uuml;m&uuml;zde bir&ccedil;ok uygulamada kullanılmaktadır. Bu &ccedil;alışmada D sınıfı rezonans evirici analizi Runge-Kutta algoritması ile ger&ccedil;ekleştirilmiştir. Yapılan analizin sonucunda evirici akımı ve gerilimi, kondansat&ouml;r gerilimi gibi devre parametrelerine ait grafikler &ccedil;izdirilmiştir. Aynı devre parametreleri MATLAB Simulink &uuml;zerinde yapılan benzetim &ccedil;alışmasında da tekrarlanarak sonu&ccedil;lar karşılaştırılmıştır ve Runge-Kutta algoritmasının doğruluğu ortaya konulmuştur.

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Wang, Fang Zong, Yi Fan He, and Jing Ye. "Transient Stability Simulation by Explicit and Symplectic Runge-Kutta-Nyström Method." Advanced Materials Research 383-390 (November 2011): 1960–64. http://dx.doi.org/10.4028/www.scientific.net/amr.383-390.1960.

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The symplectic algorithm is a kind of new numerical integration methods. This paper proposes the application of the explicit and symplectic Runge-Kutta-Nyström method to solve the differential equations encountered in the power system transient stability simulation. The proposed method achieves significant improvement both in speed and in calculation precision as compared to the conventional Runge-Kutta method which is widely used for power system transient stability simulation. The proposed method is applied to the IEEE 145-bus system and the results are reported.

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Duan, Xiaoming, Jinsong Leng, Carlo Cattani, and Caiyun Li. "A Shannon-Runge-Kutta-Gill Method for Convection-Diffusion Equations." Mathematical Problems in Engineering 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/163734.

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A Shannon-Rugge-Kutta-Gill method for solving convection-diffusion equations is discussed. This approach transforms convection-diffusion equations into one-dimensional equations at collocations points, which we solve by Runge-Kutta-Gill method. A concrete example solved is used to examine the method’s feasibility.

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Darti, Isnani. "Perambatan Gelombang Optik pada Grating Sinusoidal dengan Chirp dan Taper." CAUCHY 1, no. 1 (November 15, 2009): 31. http://dx.doi.org/10.18860/ca.v1i1.1697.

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Artikel ini membahas model perambatan gelombang optik pada grating sinusoidal takhomogen. Model tersebut diturunkan dengan mereduksi secara eksak persamaan Helmholtz menjadi sistem persamaan diferensial orde satu dengan syarat awal yang dapat diselesaikan dengan metode Runge-Kutta orde empat. Metode ini disebut Metode Integrasi<br />Langsung (MIL). Formulasi MIL sangat sederhana baik dalam hal penurunannya maupun implementasinya karena tidak memerlukan prosedur iterasi maupun optimasi. Dengan<br />menggunakan MIL, dipelajari perubahan respon optik pada grating sinusoidal akibat variasi amplitudo modulasi indeks (taper) dan variasi frekuensi spasial grating (chirp). Hasil simulasi menunjukkan bahwa taper menyebabkan adanya fenomena penghilangan side-lobe pada spektrum transmitansi. Adanya chirp menyebabkan penghalusan side-lobe pada spektrum transmitansi dengan semakin besar parameter chirp menyebabkan peningkatan transmitansi di sekitar pusat band-gap dari grating homogen. Selain implementasi integrasi numerik (Runge-Kutta), MIL merupakan metode eksak sehingga dapat digunakan untuk mengevaluasi validitas metode yang sering digunakan yaitu Persamaan Moda Tergandeng (PMT). Dari hasil perbandingan dapat disimpulkan bahwa secara umum PMT kurang akurat dalam menganalisis struktur grating sinusoidal baik homogen maupun tak-homogen.

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Arshad, Muhammad Sarmad, Dumitru Baleanu, Muhammad Bilal Riaz, and Muhammad Abbas. "A Novel 2-Stage Fractional Runge–Kutta Method for a Time-Fractional Logistic Growth Model." Discrete Dynamics in Nature and Society 2020 (June 10, 2020): 1–8. http://dx.doi.org/10.1155/2020/1020472.

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In this paper, the fractional Euler method has been studied, and the derivation of the novel 2-stage fractional Runge–Kutta (FRK) method has been presented. The proposed fractional numerical method has been implemented to find the solution of fractional differential equations. The proposed novel method will be helpful to derive the higher-order family of fractional Runge–Kutta methods. The nonlinear fractional Logistic Growth Model is solved and analyzed. The numerical results and graphs of the examples demonstrate the effectiveness of the 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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