Journal articles: 'Runge-Kutta 4th and 5th order'

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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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Raza, J., Fateh Mebarek-Oudina, Paras Ram, and S. Sharma. "MHD Flow of Non-Newtonian Molybdenum Disulfide Nanofluid in a Converging/Diverging Channel with Rosseland Radiation." Defect and Diffusion Forum 401 (May 2020): 92–106. http://dx.doi.org/10.4028/www.scientific.net/ddf.401.92.

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The steady two-dimensional flow of an incompressible non-Newtonian Molybdenum Disulfide nanofluid in the presence of source or sink between two stretchable or shrinkable walls under the influence of thermal radiation is investigated numerically. A generalized transformation is applied to convert the constructed set of partial differential equations (PDEs) into the system of non-linear coupled ordinary differential equations (ODEs). The obtained system of ODEs are solved by using Runge-Kutta 4th and 5th order. The influence of physical parameters, shrinking/ stretching parameter, Casson parameter, Hartmann number, Reynolds number, solid volume fraction, opening angle of the channel and radiation parameter on the velocity and temperature distribution are observed for converging and diverging channels. It is noticed that thermal boundary layer thickness is diminished for increased thermal radiation resulting in gradual temperature fall. The results also reveal that velocity and temperature profile both are elevated on raising the stretching parameter and Hartmann number. A comparative analysis is made out to validate the present results.

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Makukula, Z. G., S. S. Motsa, and S. Shateyi. "Numerical Analysis for the Synthesis of Biodiesel Using Spectral Relaxation Method." Mathematical Problems in Engineering 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/601374.

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Biodiesel is an alternative diesel fuel chemically defined as the mono-alkyl esters of long chain fatty acids derived from vegetable oils or animal fat. It is becoming more attractive as an alternative fuel due to the depleting fossil fuel resources. A mathematical model for the synthesis of biodiesel from vegetable oils and animal fats is presented in this study. Numerical solutions of the model are found using a spectral relaxation method. The method, originally developed for boundary value problems, is an iterative scheme based on the Chebyshev spectral collocation method developed by decoupling systems of equations using Gauss-Seidel type of techniques. The effects of the reaction rate constants and initial concentrations of the reactants on the amount of the final product are being investigated. The accuracy of the numerical results is validated by comparison with known analytical results and numerical results obtained usingode45, an efficient explicit 4th and 5th order Runge-Kutta method used to integrate both linear and nonlinear differential equations.

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Zhao, Xin, Jian Li, Wansuo Duan, and Dongqian Xue. "Numerical Analysis of the Mixed 4th-Order Runge-Kutta Scheme of Conditional Nonlinear Optimal Perturbation Approach for the EI Niño-Southern Oscillation Model." Advances in Applied Mathematics and Mechanics 8, no. 6 (September 19, 2016): 1023–35. http://dx.doi.org/10.4208/aamm.2014.m786.

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AbstractIn this paper, we proposes and analyzes the mixed 4th-order Runge-Kutta scheme of conditional nonlinear perturbation (CNOP) approach for the EI Niño-Southern Oscillation (ENSO) model. This method consists of solving the ENSO model by using a mixed 4th-order Runge-Kutta method. Convergence, the local and global truncation error of this mixed 4th-order Runge-Kutta method are proved. Furthermore, optimal control problem is developed and the gradient of the cost function is determined.

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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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Álvarez, Cristhian, Edwin Espinel, and Carlos J. Noriega. "Study of Different Alternatives for Dynamic Simulation of a Steam Generator Using MATLAB." Fluids 6, no. 5 (April 29, 2021): 175. http://dx.doi.org/10.3390/fluids6050175.

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This work presents the simulation of a steam generator or water-tube boiler through the implementation in MATLAB® for a proposed mathematical model. Mass and energy balances for the three main components of the boiler—the drum, the riser and down-comer tubes—are presented. Three alternative solutions to the ordinary differential equation (ODE) were studied, based on Runge–Kutta 4th order method, Heun’s method, and MATLAB function Ode45. The best results were obtained using MATLAB® function Ode45 based on the Runge–Kutta 4th Order Method. The error was less than 5% for the simulation of the steam pressure in the drum, the total volume of water in the boiler, and the mixture quality in relation to what was reported.

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Ashgi, Rizky. "Comparison of Numerical Simulation of Epidemiological Model between Euler Method with 4th Order Runge Kutta Method." International Journal of Global Operations Research 2, no. 1 (March 14, 2021): 37–44. http://dx.doi.org/10.47194/ijgor.v2i1.67.

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Coronavirus Disease 2019 has become global pandemic in the world. Since its appearance, many researchers in world try to understand the disease, including mathematics researchers. In mathematics, many approaches are developed to study the disease. One of them is to understand the spreading of the disease by constructing an epidemiology model. In this approach, a system of differential equations is formed to understand the spread of the disease from a population. This is achieved by using the SIR model to solve the system, two numerical methods are used, namely Euler Method and 4th order Runge-Kutta. In this paper, we study the performance and comparison of both methods in solving the model. The result in this paper that in the running process of solving it turns out that using the euler method is faster than using the 4th order Runge-Kutta method and the differences of solutions between the two methods are large.

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Randhir, D., S. Umashankar, D. Vijayakumar, and D. P. Kothari. "Comparative Analysis of Solution Methods to Power Electronic Interface Modeling for Renewable Energy Applications." Advanced Materials Research 768 (September 2013): 9–15. http://dx.doi.org/10.4028/www.scientific.net/amr.768.9.

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This paper presents the comparison between various numerical methods namely Euler's Method, Midpoint Method, 2nd order Runge Kutta Method and 4th order Runge Kutta Method with the analytical method to solve a power electronic system in both single phase and three phase configuration using decoupled methodology. The values of source current, load current and DC link voltage are obtained for each method using Matlab software and compared with each other. Also, the error in each numerical method with respect to analytical method is calculated and tabulated. These power electronic models could be an excellent research platform for testing the renewable energy systems without going for full scale or scaled down models.

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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 (May 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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ANASTASSI, Z. A., and T. E. SIMOS. "SPECIAL OPTIMIZED RUNGE–KUTTA METHODS FOR IVPs WITH OSCILLATING SOLUTIONS." International Journal of Modern Physics C 15, no. 01 (January 2004): 1–15. http://dx.doi.org/10.1142/s0129183104006510.

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In this paper we present a family of explicit Runge–Kutta methods of 5th algebraic order, one of which has variable coefficients, for the efficient solution of problems with oscillating solutions. Emphasis is placed on the phase-lag property in order to show its importance with regards to problems with oscillating solutions. Basic theory of Runge–Kutta methods, phase-lag analysis and construction of the new methods are described. Numerical results obtained for known problems show the efficiency of the new methods when they are compared with known methods in the literature. Furthermore we note that the method with variable coefficients appears to have much higher accuracy, which gets close to double precision, when the product of the frequency with the step-length approaches certain values. These values are constant and independent of the problem solved and depend only on the method used and more specifically on the expressions used to achieve higher algebraic order.

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Evans, D. J. "A new 4th order runge-kutta method for initial value problems with error control." International Journal of Computer Mathematics 39, no. 3-4 (January 1991): 217–27. http://dx.doi.org/10.1080/00207169108803994.

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Cherifi, Dalila, Messaoud Boudjada, Abdelatif Morsli, Gabriel Girard, and Rachid Deriche. "Combining Improved Euler and Runge-Kutta 4th order for Tractography in Diffusion-Weighted MRI." Biomedical Signal Processing and Control 41 (March 2018): 90–99. http://dx.doi.org/10.1016/j.bspc.2017.11.008.

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Lupu, Sergiu, and Eugen Zaharescu. "Effects Of Direct And Indirect Solar Radiation Pressure In Orbital Parameters Of GPS Satelittes." Analele Universitatii "Ovidius" Constanta - Seria Matematica 22, no. 2 (June 1, 2014): 141–50. http://dx.doi.org/10.2478/auom-2014-0039.

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AbstractIn this paper we determined the variation of the Keplerian orbital elements of a GPS satellite due to the direct and indirect action of solar radiation pressure. For this study, we created a soft program to determine the Keplerian elements. This soft uses the initial conditions of position and speed of a GPS satellite and solves the Laplace integration problem using Runge - Kutta algorithm of 4th order.

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Banshiwal, Annu, and M. Goyal. "MHD flow due to stretching sheet embedded in non Darcian porous medium with thermal stratification effects." Journal of Naval Architecture and Marine Engineering 15, no. 1 (June 28, 2018): 65–74. http://dx.doi.org/10.3329/jname.v15i1.29692.

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The aim of present study is to analyze the non- Darcian effects on unsteady non-linear MHD flow of an incompressible, electrically conducting and viscous fluid over a stretching sheet embedded in a porous medium with heat source, viscous dissipation and thermal stratification. The dimensionless governing equation solved numerically by using 4th order Runge - Kutta method. The effects of pertinent parameters on velocity and temperature depicted graphically.

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Adam, Riza Ibnu, Susilawati Susilawati, and Adhi Rizal. "VARIATIONAL MUTI-STEPS METHOD TO SOLVE DAMPED OSCILLATION EQUATION." Jurnal Neutrino 10, no. 1 (January 15, 2018): 14. http://dx.doi.org/10.18860/neu.v10i1.4399.

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This paper aims to identifying the numerical method accuracy of the analytical solution of the damped oscillation equation motion. Adams method of 4th order, Milne method and Adams-Simpson method are used to find numerical solutions. Value of y(1), y(2), y(3) obtained from The 4th order Runge-Kutta method. They used as initial value of multistep method. Then, the numerical solution result was compared with analytical solution. From the research result, it is found that 4th order Adams method has the best accuracy.

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Zhan, Xue Ping, Kuan Jun Zhu, Cao Lan Liu, Bin Liu, Jun Zhang, and Guang Rui Tang. "Research on Numerical Simulation of Quad Bundle Conductor on Galloping." Applied Mechanics and Materials 457-458 (October 2013): 23–27. http://dx.doi.org/10.4028/www.scientific.net/amm.457-458.23.

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The models of the multi-bundled conductors are constructed by finite element method in this paper. The numerical results are given by using the 4th order Runge-Kutta method considering aerodynamic parameter of sub-conductor. The simulation results are obtained on galloping of quad bundle conductors with the different span. Thus some effective numerical results of quad twin bundle conductor can provide a useful reference for anti-galloping design.

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Zhan, Xue Ping, Kuan Jun Zhu, Bin Liu, Ya Duo Liu, and Xin Min Li. "Research on Nonlinear Numerical Simulation Method of Icing Conductor on Galloping." Advanced Materials Research 706-708 (June 2013): 1799–804. http://dx.doi.org/10.4028/www.scientific.net/amr.706-708.1799.

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In this paper,the models of the conductors are constructed by finite element method with a three-node, isoparametric cable element. Two numerical simulation methods that include the 4th order Runge-Kutta (referred to as R-K) method and improved time integration method on galloping are studied, and the numerical results on galloping are given by using those two methods by programming. The study result shows that dynamic responses simulated on galloping by using 4th order R-K method and improved time integration method are in good agreement in this paper, but the efficiency of improved time integration method is higher. Result of this paper can provide beneficial reference for galloping simulation calculation.

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Fagbemi, Latif, David Adamon, and Evrard Karol Ekouedjen. "Modeling Carbon-to-Nitrogen Ratio Influence on Biogas Production by the 4th-order Runge–Kutta Method." Energy & Fuels 33, no. 9 (August 16, 2019): 8721–26. http://dx.doi.org/10.1021/acs.energyfuels.9b01721.

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Khan, Zeeshan, Haroon Ur Rasheed, Wali Khan Mashwani, Nasser Tairan, Habib Shah, Qayyum Shah, Waris Khan, and Ikram Ullah. "Investigation of Wire Coating Using Hydromagnetic Third-Grade Liquid for Coating along with Hall Current and Porous Medium." Mathematical Problems in Engineering 2020 (July 27, 2020): 1–8. http://dx.doi.org/10.1155/2020/4218717.

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In this research study, we investigated and performed coating analysis of wire by using MHD convective third-order fluid in the presence of a permeable matrix taking into account the Hall current. The equations that control the motion of fluid in the chamber are first modeled and then numerically solved by using 4th order Runge–Kutta–Fehlberg technique. The Runge–Kutta–Fehlberg method is a powerful tool used in this article to attain a numerical solution for a system of nonlinear ordinary differential equations describing the problem of fluid flow. The impact of governing parameters on velocity and temperature profiles is investigated graphically. It is noticed that the velocity profiles ur rise as the value of viscoelastic parameter β increases and slow down when the permeability parameter K and the Hartmann number M increase. Also, the temperature profiles θr enhance as the Brinkman number Br, permeability parameter K, magnetic parameter M, and non-Newtonian parameter β increase. For the sake of validation, the proposed method is also compared with BVPh2, and good agreement is found. Furthermore, a comparison is also done with the published work as a limiting case.

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Hieu, D. V., N. Q. Hai, and D. T. Hung. "The Equivalent Linearization Method with a Weighted Averaging for Solving Undamped Nonlinear Oscillators." Journal of Applied Mathematics 2018 (2018): 1–15. http://dx.doi.org/10.1155/2018/7487851.

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The Equivalent Linearization Method (ELM) with a weighted averaging is applied to analyze five undamped oscillator systems with nonlinearities. The results obtained via this method are compared with the ones achieved by Parameterized Perturbation Method (PPM), Min–Max Approach (MMA), Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Energy Balance Method (EBM), Harmonic Balance Method (HBM), 4th-Order Runge-Kutta Method, and the exact ones. The obtained results demonstrate that this method is very convenient for solving nonlinear equations and also can be successfully applied to a lot of practical engineering and physical problems.

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Hasanuddin, Hasanuddin. "Analisis Galat Energi dan Galat Fase Metode Forward 4th Order Symplectic Chin-Chen untuk Kasus Sistem Osilator Harmonik Sederhana." POSITRON 10, no. 2 (December 29, 2020): 9. http://dx.doi.org/10.26418/positron.v10i2.40023.

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Penelitian ini meninjau galat energi dan galat fase dari metode forward 4th order symplectic Chin-Chen (F4OS-CC) untuk kasus sistem osilator harmonik sederhana. Metode F4OS-CC merupakan suatu integrator yang bersifat symplectic dan time-reversible tanpa menggunakan selang waktu negatif. Simulasi osilasi harmonik sederhana dengan metode integrasi F4OS-CC dan metode Leapfrog menghasilkan galat energi terikat (tidak memiliki rambatan linier). Akan tetapi, metode F4OS-CC memiliki galat fase yang lebih besar jika dibandingkan dengan metode non-symplectic orde sama (metode Runge-Kutta orde 4). Galat fase dapat dikurangi dengan menggunakan selang waktu variabel yang bergantung terhadap posisi. Tetapi, rambatan galat energi menjadi meningkat secara linier.

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Sarsembayev, M., B. Urmashev, O. Mamyrbayev, M. Turdalyuly, and T. Sarsembayeva. "USING THE CUDA TECHNOLOGY TO SPEED UP COMPUTATIONS IN PROBLEMS OF CHEMICAL KINETICS." PHYSICO-MATHEMATICAL SERIES 2, no. 336 (April 15, 2021): 39–47. http://dx.doi.org/10.32014/2021.2518-1726.19.

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The main idea of the implementation is reducing the time for calculation and thereby implement a multi-user mode for users by placing it on a server with access via a web browser. To model the kinetics of chemical reacting systems were used 4th and 5th grade Runge-Kutta methods and to receive the index of advantages of this elaboration were written programs in C# for sequential computation on a central processor and was used a platform for parallel computation of CUDA on graphic processors. Parallelization of data during calculation on a GPU was performed by the distribution of the reaction to individual strands, when changes of the concentration was calculated over a given time interval of a certain substance. Parallelization is performed over all elementary reactions, with the increasing of the number of reactions in the mechanism, because of this the computation on the GPU has a noticeable gain in time.

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Zhan, Xue Ping, Ya Duo Liu, Bin Liu, and Kuan Jun Zhu. "Nonlinear Numerical Simulation on Galloping of Twin Bundle Conductor Considering the Wake Effect." Advanced Materials Research 614-615 (December 2012): 1390–93. http://dx.doi.org/10.4028/www.scientific.net/amr.614-615.1390.

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In this paper, the models of the multi-bundled conductors are constructed by finite element method. The wake effect of aerodynamic parameter of sub-conductor of the windward side relative to the leeward side is studied. The numerical results are given by using the 4th order Runge-Kutta method. Similarly, the proposed model can be added to the different aerodynamic loads on each individual sub-conductor of a bundle conductor during the simulation of galloping. Thus the numerical results are much closer to the actual movement of galloping and provide a useful reference for anti-galloping.

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Dietiker, J.-F., K. A. Hoffmann, M. Papadakis, and R. Agarwal. "Development of Three-Dimensional PML Boundary Conditions for Aeroacoustics Applications." International Journal of Aeroacoustics 1, no. 3 (September 2002): 307–27. http://dx.doi.org/10.1260/147547202320962600.

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Perfectly Matched Layer (PML) boundary conditions are derived in generalized curvilinear coordinates for three-dimensional aeroacoustic applications. The resulting governing equations are solved numerically by a four-stage Runge-Kutta scheme, with 4th/6th order compact finite difference formulation. The PML equations are programmed in a subroutine, which is easily incorporated to the main program LINEULER (Linearized Euler's equation solver). Two and three-dimensional benchmarks problems are solved to investigate the efficiency and accuracy of the PML boundary conditions. Investigations on the PML parameters have been conducted to determine the optimum combination of parameters used in the computations.

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Zainol, Norazwina, and Siti Natrah Ismail. "Evaluation of Enzyme Kinetic Parameters to Produce Methanol Using Michaelis-Menten Equation." Bulletin of Chemical Reaction Engineering & Catalysis 14, no. 2 (August 1, 2019): 436. http://dx.doi.org/10.9767/bcrec.14.2.3317.436-442.

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Determination of kinetic parameters of enzymes is important in biotechnology research. It is also one of the most challenging processes in methanol production. The activity of enzyme is determined in term of initial rates at various substrate concentrations. The enzymatic hydrolysis of methanol by pectin methyl esterase (PME) enzyme was investigated at 25 °C and pH 9 over the reaction time range from 0 to 90 min. In this study, the parameters of the enzyme's kinetic, KM and Vmax were directly determined using a modified Michaelis-Menten equation by applying the Lineweaver-Burk plots. Besides, nonlinear regression of Michaelis-Menten equation was calculated based on Euler’s and Runge-Kutta 4th order methods by using Solver supplement application. The result of kinetic constant was tested by comparing the experimental data with model predictions. It was found that Euler and Runge-Kutta method was successful in determining the kinetic parameter rather than Lineweaver-Burk plot. The application of the Michaelis-Menten equation describes the enzyme kinetic very well. From the kinetic analysis, it showed the good agreement between the result obtained and the predictions model in the production of methanol using PME enzyme. Copyright © 2019 BCREC Group. All rights reserved

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Resmawan, M. Eka, Nurwan, and N. Achmad. "Analisis Kontrol Optimal Pada Model Matematika Penyebaran Pengguna Narkoba Dengan Faktor Edukasi." JURNAL ILMIAH MATEMATIKA DAN TERAPAN 17, no. 2 (November 21, 2020): 238–48. http://dx.doi.org/10.22487/2540766x.2020.v17.i2.15201.

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ABSTRACT This paper discusses the mathematical model of drug users with education. Optimal control theory was used on this model with education as a control to achieve the goal of minimizing the number of drug users. The optimal control problem was analyzed using Pontryagin’s minimum principle and performed numerical simulation by using a 4th-order Runge-Kutta method. Based on the numerical simulation, there was a change in the number in each population which caused the population with education to increase, and control with education resulted in the reduced number of drug users. Keywords: Optimal control; mathematical model; drug users; education ABSTRAK Artikel ini membahas tentang model matematika penyebaran pengguna narkoba dengan faktor edukasi. Teori kontrol optimal diterapkan pada model ini dengan pemberian kontrol berupa edukasi dengan tujuan untuk meminimumkan jumlah pengguna narkoba. Kontrol optimal dianalisis menggunakan Prinsip Minimum Pontryagin dan dilakukan simulasi numerik dengan menggunakan metode Runge-Kutta orde 4. Berdasarkan simulasi diperoleh bahwa terjadi perubahan jumlah di tiap populasi dan mengakibatkan jumlah populasi dengan edukasi bertambah, serta pemberian kontrol dengan edukasi mengakibatkan jumlah pengguna narkoba berkurang. Kata kunci : Kontrol optimal; model matematika; pengguna narkoba; edukasi

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Kadirbayeva, Zh M., E. A. Bakirova, A. Sh Dauletbayeva, and A. A. Kassymgali. "AN ALGORITHM FOR SOLVING A BOUNDARY VALUE PROBLEM FOR ESSENTIALLY LOADED DIFFERENTIAL EQUATIONS." PHYSICO-MATHEMATICAL SERIES 2, no. 336 (April 15, 2021): 6–14. http://dx.doi.org/10.32014/2021.2518-1726.15.

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A linear boundary value problem for essentially loaded differential equations is considered. Using the properties of essentially loaded differential we reduce the considering problem to a two-point boundary value problem for loaded differential equations. This problem is investigated by parameterization method. We offer algorithm for solving to boundary value problem for the system of loaded differential equations. This algorithm includes of the numerical solving of the Cauchy problems for system of the ordinary differential equations and solving of the linear system of algebraic equations. For numerical solving of the Cauchy problem we apply the Runge–Kutta method of 4th order. The proposed numerical implementation is illustrated by example.

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Mohammed, Mokhtar, and Taha Janan Mourad. "Theoretical Analysis of a New Design of a Concentration Based Solar Distiller." E3S Web of Conferences 234 (2021): 00003. http://dx.doi.org/10.1051/e3sconf/202123400003.

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This paper includes a theoretical study of energy balance for all parts of new design of solar concentration distiller using a parabolic concentrator with a half-cylinder basin. Our goal is to analyze the thermal efficiency of the new device to use in the Morocco's Rabat-Sale-Kenitra region. The methodology concentrates on solving the thermal collector's energy balance equations whose components are the glass cover, the brackish water and half-cylinder absorber. Numerical resolution of the energy balance equations was performed using a MATLAB code based on the method of 4th order Runge-Kutta. The results show a good theoretical performance of the new device.

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Sharma, R. P., Seema Tinker, B. J. Gireesha, and B. Nagaraja. "Effect of Convective Heat and Mass Conditions in Magnetohydrodynamic Boundary Layer Flow with Joule Heating and Thermal Radiation." International Journal of Applied Mechanics and Engineering 25, no. 3 (September 1, 2020): 103–16. http://dx.doi.org/10.2478/ijame-2020-0037.

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AbstractA free convection viscous MHD flow over a semi-infinite vertical sheet with convective heat and mass conditions has been considered. The effects of thermal radiation, chemical reaction and Joule heating on flow are also accounted. The governing nonlinear partial differential equations have been transformed into a set of highly non-linear coupled ordinary differential equations (ODEs) using appropriate similarity transformations. Numerical solutions of transformed equations are obtained by employing the 5th order Runge-Kutta Fehlberg technique followed by the shooting technique. The influences of different flow parameters on the momentum, energy and mass field are discussed and shown graphically. Results reveal that temperature and concentration profiles enhance due to increasing heat and mass Biot number parameters.

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Ji, Zhen-Gang, Walter R. Johnson, James M. Price, and Charles F. Marshall. "Oil-Spill Risk Analysis for Assessing Environmental Impacts." International Oil Spill Conference Proceedings 2003, no. 1 (April 1, 2003): 1125–29. http://dx.doi.org/10.7901/2169-3358-2003-1-1125.

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ABSTRACT The Minerals Management Service (MMS), an agency of the U.S. Department of the Interior, maintains a leasing program for commercial oil and gas development on the Outer Continental Shelf in U.S. territorial waters. The MMS performs an oil-spill risk analysis (OSRA) using, in part, a statistical model of hypothetical oil-spill trajectories. The OSRA Model is driven by analyzed sea surface winds and model-generated ocean surface currents. Instead of focusing on individual oil-spill events, the OSRA examines oil-spill risks over long periods of time, ranging from 5 years to decades. The OSRA Model calculates thousands of oil-spill trajectories over extended areas of the U. S. continental shelf and tabulates the frequencies with which the simulated oil-spills contact the geographic boundaries of designated natural resources within a specified number of days after the simulated spill events. A key element of OSRA Model runs is the particle trajectory simulation based on wind velocities and surface ocean currents. It is critical that the trajectory model has a numerical scheme that is stable, accurate, and efficient. Currently, two numerical schemes are incorporated into the model: the Eulerian scheme and the 4th order Runge-Kutta scheme. To test the numerical schemes, a set of analytical solutions to the Navier-Stokes equations is developed. The analytical velocities and particle trajectories are compared with the numerical solutions from the two numerical schemes. The stability and efficiency of the two schemes are discussed. Applying the model to the Gulf of Mexico using 9 years of winds and ocean currents, we find that the OSRA Model with the 4th order Runge-Kutta scheme is much more efficient and able to better represent the circulation patterns and particle movements in the Gulf of Mexico.

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Alwaan, Ismaeel M. "Experimental and Runge–Kutta Method Simulation to Investigate Corrosion Kinetics of Mild Steel in Sulfuric Acid Solutions." International Journal of Corrosion 2018 (July 8, 2018): 1–6. http://dx.doi.org/10.1155/2018/9087101.

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The mild steel is extensively used in different industrial applications and the biggest problem in the application of mild steel is corrosion. In this work, the reaction kinetics of mild steel with sulfuric acid at different concentrations and at different temperatures were studied in combination with the experimental data and theoretical approach using the Runge–Kutta method. The results revealed that the rate of reaction constant for temperatures in the range of 30–50°C was changed from 2618 to 2793 L3/mol3.h, respectively. The order of reaction of mild steel was 4th order in all temperature ranges. The enthalpy, entropy, and Gibbs free energy of mild steel reaction at a temperature of 298 K were estimated. The activation energy (E/R) of the reaction was 4.829 K. It was concluded that the sulfuric acid reaction with mild steel occurred easily and the inhibitors should be used in these systems.

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Fragkou, Anastasia, and Panos Papanicolaou. "Positively and Negatively Round Turbulent Buoyant Jets into Homogeneous Calm Ambient." Proceedings 2, no. 11 (July 31, 2018): 572. http://dx.doi.org/10.3390/proceedings2110572.

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A mathematical model has been employed to determine the characteristics of Boussinesq round buoyant jets which are injected horizontally or at an angle to horizontal, into a homogeneous, calm ambient. The solution of a system of three conservation first order nonlinear differential equations was obtained with a 4th Runge-Kutta scheme, using an entrainment coefficient which is related to the local Richardson number of the flow. Two types of positively and negatively buoyant jets were investigated (i) those where the buoyancy is a function of salinity henceforth called saline jets, and (ii) those where the buoyancy is a function of the temperature difference between jet and ambient fluid, henceforth called thermal jets.

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Ga, Bui Van, Nhan Hong Quang, and Jean Marc Vignon. "Calculation of turbulent diffusion jets under effects of gravity and moving surrounding air." Vietnam Journal of Mechanics 23, no. 2 (June 1, 2001): 87–94. http://dx.doi.org/10.15625/0866-7136/9942.

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The basis theory for the turbulent diffusion of jet and flame has been presented previously [1, 2]. But that one applies only in quiet surrounding air with the effects of buoyancy neglected. In the present paper, the theory is developed further by establishing an integral model for a jet in more general conditions with variable inclined angles, under effects of gravity and surrounding air velocity in any direction compared to the jet axis. The system of equations is closed by turbulence k-E model and is solved by 4th order Runge-Kutta method. In the first stage, the model is applied to predict the velocity field, the concentration field and with development of a 0.3 m diameter jet.

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Shahrir, Mohammad Shazri, N. Kumaresan, Kuru Ratnavelu, and M. Z. M. Kamali. "A Multiagent Transfer Function Neuroapproach to Solve Fuzzy Riccati Differential Equations." Journal of Applied Mathematics 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/605625.

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A numerical solution of fuzzy quadratic Riccati differential equation is estimated using a proposed new approach for neural networks (NN). This proposed new approach provides different degrees of polynomial subspaces for each of the transfer function. This multitude of transfer functions creates unique “agents” in the structure of the NN. Hence it is named as multiagent neuroapproach (multiagent NN). Previous works have shown that results using Runge-Kutta 4th order (RK4) are reliable. The results can be achieved by solving the 1st order nonlinear differential equation (ODE) that is found commonly in Riccati differential equation. Multiagent NN shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over Mabood et al. (2013), RK-4, and the existing neuromethod (NM). Numerical examples are discussed to illustrate the proposed method.

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35

Ali Lund, Liaquat, Zurni Omar, Ilyas Khan, Jawad Raza, Mohsen Bakouri, and I. Tlili. "Stability Analysis of Darcy-Forchheimer Flow of Casson Type Nanofluid Over an Exponential Sheet: Investigation of Critical Points." Symmetry 11, no. 3 (March 20, 2019): 412. http://dx.doi.org/10.3390/sym11030412.

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In this paper, steady two-dimensional laminar incompressible magnetohydrodynamic flow over an exponentially shrinking sheet with the effects of slip conditions and viscous dissipation is examined. An extended Darcy Forchheimer model was considered to observe the porous medium embedded in a non-Newtonian-Casson-type nanofluid. The governing equations were converted into nonlinear ordinary differential equations using an exponential similarity transformation. The resultant equations for the boundary values problem (BVPs) were reduced to initial values problems (IVPs) and then shooting and Fourth Order Runge-Kutta method (RK-4th method) were applied to obtain numerical solutions. The results reveal that multiple solutions occur only for the high suction case. The results of the stability analysis showed that the first (second) solution is physically reliable (unreliable) and stable (unstable).

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Keppel, W. J., and D. A. DaDeppo. "Finite Axisymmetric Deformation of Rubber-Like Shells of Revolution." Journal of Applied Mechanics 55, no. 2 (June 1, 1988): 332–40. http://dx.doi.org/10.1115/1.3173679.

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The finite axisymmetric deformation of a thin shell of revolution is treated in this analysis. The governing differential equations are given for a hyperelastic shell material with the Mooney-Rivlin strain-energy-density function. These equations are solved numerically using a 4th-order Runge-Kutta integration method. A generalized Newton-Raphson iteration procedure is used to systematically improve trial solutions of the differential equations. The governing differential equations are differentiated with respect to a set of generalized coordinates to derive associated rate equations. The rate equations are solved numerically to generate the tangent stiffness matrix which is used to determine the load deformation history of the shell with incremental loading. Numerical examples are presented to illustrate the major characteristics of nonlinear shell behavior.

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Chuathong, Nissaya, and Sayan Kaennakham. "Numerical Solution to Coupled Burgers’ Equations by Gaussian-Based Hermite Collocation Scheme." Journal of Applied Mathematics 2018 (September 27, 2018): 1–18. http://dx.doi.org/10.1155/2018/3416860.

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One of the most challenging PDE forms in fluid dynamics namely Burgers equations is solved numerically in this work. Its transient, nonlinear, and coupling structure are carefully treated. The Hermite type of collocation mesh-free method is applied to the spatial terms and the 4th-order Runge Kutta is adopted to discretize the governing equations in time. The method is applied in conjunction with the Gaussian radial basis function. The effect of viscous force at high Reynolds number up to 1,300 is investigated using the method. For the purpose of validation, a conventional global collocation scheme (also known as “Kansa” method) is applied parallelly. Solutions obtained are validated against the exact solution and also with some other numerical works available in literature when possible.

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Djakaria, Ismail, Muhammad Bachtiar Gaib, and Resmawan Resmawan. "Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behavior." CAUCHY 6, no. 4 (May 30, 2021): 260–69. http://dx.doi.org/10.18860/ca.v6i4.11472.

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This paper discusses the analysis of the Rosenzweig-MacArthur predator-prey model with anti-predator behavior. The analysis is started by determining the equilibrium points, existence, and conditions of the stability. Identifying the type of Hopf bifurcation by using the divergence criterion. It has shown that the model has three equilibrium points, i.e., the extinction of population equilibrium point (E0), the non-predatory equilibrium point (E1), and the co-existence equilibrium point (E2). The existence and stability of each equilibrium point can be shown by satisfying several conditions of parameters. The divergence criterion indicates the existence of the supercritical Hopf-bifurcation around the equilibrium point E2. Finally, our model's dynamics population is confirmed by our numerical simulations by using the 4th-order Runge-Kutta methods.

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Hao, Y. X., W. Zhang, and X. L. Ji. "Nonlinear Dynamic Response of Functionally Graded Rectangular Plates under Different Internal Resonances." Mathematical Problems in Engineering 2010 (2010): 1–12. http://dx.doi.org/10.1155/2010/738648.

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The nonlinear dynamic response of functionally graded rectangular plates under combined transverse and in-plane excitations is investigated under the conditions of 1 : 1, 1 : 2 and 1 : 3 internal resonance. The material properties are assumed to be temperature-dependent and vary along the thickness direction. The thermal effect due to one-dimensional temperature gradient is included in the analysis. The governing equations of motion for FGM rectangular plates are derived by using Reddy's third-order plate theory and Hamilton's principle. Galerkin's approach is utilized to reduce the governing differential equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms, which are then solved numerically by using 4th-order Runge-Kutta algorithm. The effects of in-plane excitations on the internal resonance relationship and nonlinear dynamic response of FGM plates are studied.

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INEL, F., Y. ZENNIR, and R. ADLENE. "Design and Simulation of Graphical User interface for Cables Parallel Robot." Algerian Journal of Signals and Systems 5, no. 1 (March 15, 2020): 66–72. http://dx.doi.org/10.51485/ajss.v5i1.98.

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In this paper, we have presented the 3D cables parallel robot in pyramid’s form. The dynamic equation has been established including the dynamic behavior, in this context; we investigated to use the Runge Kutta method of 4th order to solving non-linear partial differential equation of our system. The main contribution of this work is firstly: a graphical user interface (GUI) has been developed and implemented based on the geometric model, in order to visualizing the position of end effect or, taking account model uncertainties and payload variation. Secondly, we have studied and solved our system’s dynamical equation with implementation the PD control. This last, applied for different trajectories so as to test the accurate tracking the desired trajectory simulation using MATLAB software. As a result, the simulation tests on this robot verify the efficiency performance of the proposed controller.

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Ning, De-Zhi, Wei-Hua Song, Yu-Long Liu, and Bin Teng. "A Boundary Element Investigation of Liquid Sloshing in Coupled Horizontal and Vertical Excitation." Journal of Applied Mathematics 2012 (2012): 1–20. http://dx.doi.org/10.1155/2012/340640.

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Sloshing flows in a two-dimensional rigid rectangular tank under specified excitations in the coupled horizontal and vertical modes are simulated by using a higher-order boundary element method (BEM). The liquid sloshing is formulated as an initial-boundary-value problem based on the fully nonlinear potential flow theory. And a semi-mixed Eulerian-Lagrangian technique combined with the 4th-order Runge-Kutta scheme is employed to advance the solutions in the time marching process. A smoothing technique is applied to the free surface at every several time steps to avoid the possible numerical instabilities. Numerical results obtained are compared with the available solutions to validate the developed model. The parametric studies are carried out to show the liquid sloshing effects in terms of the slosh frequencies and excitation amplitudes in horizontal and vertical modes, the second-order resonance frequency, a bottom-mounted vertical rigid baffle, free surface displacement, and hydrodynamic forces acting on the tank.

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Slashchov, Anton, and Oleksii Yalanskyi. "Substantiation of fuzzy logic algorithms for control problems of a geotechnical systems." E3S Web of Conferences 109 (2019): 00091. http://dx.doi.org/10.1051/e3sconf/201910900091.

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The article presents results of study of intelligent fuzzy logic algorithms developed on the basis of fuzzy logic methods for information system of the mine safety system. In order to prevent emergency situations caused by the lost geotechnical system stability due to the uncertain behavior of the rock mass, a new fuzzy controller was designed which could generate an additional control signal. For the fuzzy controller, methods of data fuzziness, inference and de-fuzziness were validate, and linguistic rules were designed in order to control parameters of the geotechnical system. With the help of the Cauchy problem solved by Runge-Kutta method of the 4th order, designed a software model of the proposed system which simulated the system operation. The model has proved operability and static stability of the developed algorithms. Output signal of the fuzzy controller can be used as information for estimating risk for geotechnical systems, preventing possible emergency situations and, consequently, can improve job safety in the mines.

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43

Kairi, Rishi. "Free convection around a slender paraboloid of non-Newtonian fluid in a porous medium." Thermal Science 23, no. 5 Part B (2019): 3067–74. http://dx.doi.org/10.2298/tsci170809005k.

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This paper emphasizes the radiative heat transfer of non-Newtonian fluid on free convection around a slender paraboloid in a non-Darcy porous medium. The Ostwald-de Waele power-law representation is employed to express the non-Newto?nian behavior of fluid. Similarity analysis is applied to transform the set of non-dimensional PDE into set of ODE and then the resulting system of equations are solved by 4th order Runge-Kutta scheme with Shooting technique. The control of pertinent parameters on velocity, temperature and non-dimensional heat transfer rates are analyzed through graphical representation and explored in detail. It is evident that as the radius of the slender body increases the heat transfer coefficient decreases but the role of radiation on heat transfer rate getting reduced for all feasible values of the power-law index parameter.

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Seth, Gauri Shenkar, Rohit Sharma, B. Kumbhakar, and R. Tripathi. "MHD Stagnation Point Flow over Exponentially Stretching Sheet with Exponentially Moving Free-Stream, Viscous Dissipation, Thermal Radiation and Non-Uniform Heat Source/Sink." Diffusion Foundations 11 (August 2017): 182–90. http://dx.doi.org/10.4028/www.scientific.net/df.11.182.

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An investigation is carried out for the steady, two dimensional stagnation point flow of a viscous, incompressible, electrically conducting, optically thick heat radiating fluid taking viscous dissipation into account over an exponentially stretching non-isothermal sheet with exponentially moving free-stream in the presence of uniform transverse magnetic field and non-uniform heat source/sink. The governing boundary layer equations are transformed into highly nonlinear ordinary differential equations using suitable similarity transform. Resulting boundary value problem is solved numerically with the help of 4th-order Runge-Kutta Gill method along with shooting technique. Effects of various pertinent flow parameters on the velocity, temperature field, skin friction and Nusselt number are described through figures and tables. Also, the present numerical results are compared with the earlier published results for some reduced case and a good agreement has been found among those results.

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Shao, Mingyue, Jimei Wu, Yan Wang, and Qiumin Wu. "Nonlinear Parametric Vibration and Chaotic Behaviors of an Axially Accelerating Moving Membrane." Shock and Vibration 2019 (March 11, 2019): 1–11. http://dx.doi.org/10.1155/2019/6294814.

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Nonlinear vibration characteristics of a moving membrane with variable velocity have been examined. The velocity is presumed as harmonic change that takes place over uniform average speed, and the nonlinear vibration equation of the axially moving membrane is inferred according to the D’Alembert principle and the von Kármán nonlinear thin plate theory. The Galerkin method is employed for discretizing the vibration partial differential equations. However, the solutions concerning to differential equations are determined through the 4th order Runge–Kutta technique. The results of mean velocity, velocity variation amplitude, and aspect ratio on nonlinear vibration of moving membranes are emphasized. The phase-plane diagrams, time histories, bifurcation graphs, and Poincaré maps are obtained; besides that, the stability regions and chaotic regions of membranes are also obtained. This paper gives a theoretical foundation for enhancing the dynamic behavior and stability of moving membranes.

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Kolukula, Siva Srinivas, and P. Chellapandi. "Dynamic Stability of Plane Free Surface of Liquid in Axisymmetric Tanks." Advances in Acoustics and Vibration 2013 (July 28, 2013): 1–16. http://dx.doi.org/10.1155/2013/298458.

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When liquid filled containers are excited vertically, it is known that, for some combinations of frequency and amplitude, the free surface undergoes unbounded motion leading to instability, called parametric instability or parametric resonance, while for other combinations the free surface remains plane. In this paper, the stability of the plane free surface is investigated theoretically when the vessel is a vertical axisymmetric container. The effect of coupled horizontal excitation on the stability is examined. The dynamics of sloshing flows under specified excitations are simulated numerically using fully nonlinear finite element method based on non-linear potential flow theory. A mixed Eulerian-Lagrangian technique combined with 4th-order Runge-Kutta method is employed to advance the solution in time. A regridding technique based on cubic spline is applied to the free surface for every finite time step to avoid possible numerical instabilities.

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BALAMURUGAN, PASUPATHY, and ANNAMALAI MANI. "NUMERICAL STUDIES ON VERTICAL TUBULAR GENERATOR IN VAPOR ABSORPTION REFRIGERATION SYSTEM." International Journal of Air-Conditioning and Refrigeration 19, no. 02 (June 2011): 121–29. http://dx.doi.org/10.1142/s2010132511000454.

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A model has been developed based on Colburn–Drew type formulation to analyze a vertical tube in tube stainless steel generator with forced convective boiling. Desorption of refrigerant vapor from refrigerant–absorbent solution takes place in the inner tube of the generator, when hot water through the annulus is used as heating medium. Simultaneous heat and mass transfer phenomena of desorption are described mathematically using the mass and energy balances, considering the heat and mass transfer resistances in liquid as well as vapor phases. Model equations are solved simultaneously by means of initial value problem solvers using explicit Runge–Kutta method with 4th order accuracy. A computer code has been developed in MATLAB to obtain the results. A parametric analysis has also been performed to study the effect of various parameters on the performance of the generator.

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Boreyri, Sayyad, Mohamad Javad Ketabdari, Pouya Mohtat, and Ali Moosavi. "Transverse vibration analysis of FGM plates with in-plane exponentially non-homogeneous material." International Journal of Physical Research 4, no. 2 (June 21, 2016): 43. http://dx.doi.org/10.14419/ijpr.v4i2.6045.

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In this research, free vibration of rectangular functionally graded (FG) plates with in-plane exponentially non-homogeneous material is investigated. Young’s modulus and mass density are assumed to vary between a metal-rich and a ceramic-rich zone along one in-plane direction of the plate.The governing differential equation is derived for the case, and a truncated Taylor series expansion technique is utilized to calculate natural frequencies. A Levy-type solution is obtained for plates having two simply supported edges parallel with the material gradient direction. Results for normalized natural frequency are compared with the 4th order Runge-Kutta method, and when possible with exact solution, showing an accurate agreement. Furthermore, a comprehensive parametric study is carried out to determine the effects of different boundary conditions, aspect ratios, and material variations on the free vibration of FGM plates.

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Tripathi, R. "Marangoni convection in the transient flow of hybrid nanoliquid thin film over a radially stretching disk." Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering 235, no. 4 (April 12, 2021): 800–811. http://dx.doi.org/10.1177/09544089211008052.

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Within a magnetohydrodynamic environment, Marangoni convection (Thermocapilarity effect) in an unsteady thin film of hybrid nanoliquid flow over a disk has been discussed. A set of simplified Navier-Stokes equation using boundary layer theory is written in order to model the above mentioned flow situation. The dissipative effects caused by viscosity and magnetic field have been incorporated in temperature-balance equation. A suitable choice of transform variables facilitate a system of ordinary differential equations (ODEs) from original partial differential equations (PDEs) representing the flow phenomena. This system of ODEs are solved by shooting technique in conjunction with Runge-Kutta 4th order numerical scheme. This study reveals that by increasing the surface tension along the liquid-air interface, the velocity of hybrid nanoliquid can be increased. In the context of this research work, the hybrid nanoliquid prepared by dispersing blade shaped [Formula: see text] and Cu nanoparticles, is an ideal liquid as far as liquid coolants are concerned.

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Pai, Tzu-Yi, Huang-Mu Lo, Terng-Jou Wan, Ya-Hsuan Wang, Yun-Hsin Cheng, Meng-Hung Tsai, Hsuan Tang, Yu-Xiang Sun, Wei-Cheng Chen, and Yi-Ping Lin. "A Sewer Dynamic Model for Simulating Reaction Rates of Different Compounds in Urban Sewer Pipe." Water 13, no. 11 (June 3, 2021): 1580. http://dx.doi.org/10.3390/w13111580.

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A sewer dynamic model (SDM), an innovative use of combined models, was established to describe the reactions of compounds in a pilot sewer pipe. The set of ordinary differential equations in the SDM was solved simultaneously using the fourth-order Runge–Kutta algorithm. The SDM was validated by calculating the consistency between the simulation and observation values. After the SDM was validated, the reaction rate was analyzed. For heterotrophs in the water phase and biofilm, their growth rates were greater than the organism decay rate. For ammonia, the supply rate was greater than the consumption rate at the initial time, but the supply rate was smaller than the consumption rate from the 3rd hour. The supply rate was smaller than the consumption rate for the other six compounds. The supply rate of oxygen was smaller than the consumption rate before the 4th hour because of the microorganism activities, and, subsequently, the supply rate was greater than the consumption rate after the 4th hour because of reaeration. The results of this study provide an insight into the reaction rates of different compounds in urban sewer pipes and an urban water network modeling reference for policymaking and regulation.

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Journal articles on the topic 'Runge-Kutta 4th and 5th order'

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