Relevant bibliographies by topics / Embedded Runge-Kutta-Fehlberg method

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Academic literature on the topic 'Embedded Runge-Kutta-Fehlberg method'

Author: Grafiati
Published: 3 June 2025
Last updated: 22 June 2025

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Journal articles on the topic "Embedded Runge-Kutta-Fehlberg method"

1

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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2

Kumar, K. Ganesh, B. J. Gireesha, N. G. Rudraswamy, and M. R. Krishnamurthy. "An Unsteady Flow and Melting Heat Transfer of a Nanofluid Over a Stretching Sheet Embedded in a Porous Medium." International Journal of Applied Mechanics and Engineering 24, no. 2 (2019): 245–58. http://dx.doi.org/10.2478/ijame-2019-0016.

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Abstract An unsteady flow and melting heat transfer of a nanofluid over a stretching sheet was numerically studied by considering the effect of chemical reaction and thermal radiation. The governing non-linear partial differential equations describing the flow problem are reduced to a system of non-linear ordinary differential equations using the similarity transformations and solved numerically using the Runge–Kutta–Fehlberg fourth–fifth order method. Numerical results for concentration, temperature and velocity profiles are shown graphically and discussed for different physical parameters. Effect of pertinent parameters on momentum, temperature and concentration profiles along with local Sherwood number, local skin-friction coefficient and local Nusselt number are well tabulated and discussed.
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3

Новиков, Е. А. "A variable structure algorithm using the (3,2)-scheme and the Fehlberg method." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 3 (September 15, 2015): 446–55. http://dx.doi.org/10.26089/nummet.v16r342.

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Построен (3,2)-метод третьего порядка с замораживанием матрицы Якоби, в котором $L$-устойчивыми являются основная и промежуточные численные схемы. Получено неравенство для контроля точности вычислений с использованием вложенного метода второго порядка. Предложено неравенство для контроля устойчивости явного трехстадийного метода Рунге-Кутта-Фельберга третьего порядка. Сформулирован алгоритм переменной структуры, в котором на каждом шаге явный или $L$-устойчивый метод выбираются по критерию устойчивости. Приведены результаты расчетов. A third-order (3,2)-method allowing freezing the Jacobi matrix is constructed. Its main and intermediate numerical schemes are $L$-stable. An accuracy control inequality is obtained using an embedded method of second order. A stability control inequality for the explicit three-stage Runge-Kutta-Fehlberg method of third order is proposed. A variable structure algorithm is formulated. An explicit or $L$-stable method is chosen according to the stability criterion at each step. Numerical results are discussed.
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4

Sobamowo, MG, A. A. Yinusa, AT Akinshilo, and ST Aladenusi. ""Homotopy analysis method to MHD-slip flow of an upper- convected maxwell viscoelastic nanofluid in a permeable channel embedded in a porous medium"." International Journal of Petrochemical Science & Engineering 5, no. 1 (2020): 11–20. http://dx.doi.org/10.15406/ipcse.2020.05.00118.

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The expanding applications of viscoelastic fluids in biomedical engineering and industrial processes require proper under and physical insights into the flow phenomena of the fluids. In this work, simultaneous effects of slip and magnetic field on the flow of an upper convected Maxwell (UCM) nano fluid through a permeable micro channel embedded in porous medium are analyzed using homotopy analysis method. The results of the approximate analytical solution depict very good agreements with the results of the fifth-order Runge-Kutta Fehlberg method (Cash-Karp Runge-Kutta) coupled with shooting methodfor the verification of the mathematical method used in analyzing the flow. Thereafter, the obtained analytical solutions are used to investigate the effects of pertinent rheological parameters on flow. It is observed from the results that increase in slip parameter, nano particle concentration and Darcy number lead to increase in the velocity of the upper-convected Maxwell fluid while increase in Deborah’s, Hartmann, and Reynold numbers decrease the fluid flow velocity towards the lower plate but as the upper plate is approached a reverse trend is observed. The study can be used to advance the application of upper convected Maxwell flow in the areas of in biomedical, geophysical and astrophysics.
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5

Mallikarjuna, H. B., M. C. Jayaprakash, and Raghavendra Mishra. "Three-Dimensional Boundary layer Flow and Heat Transfer of a Fluid Particle Suspension over a Stretching Sheet Embedded in a Porous Medium." Nonlinear Engineering 8, no. 1 (2019): 734–43. http://dx.doi.org/10.1515/nleng-2018-0008.

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Abstract This article presents the effect of nonlinear thermal radiation on three dimensional flow and heat transfer of fluid particle suspension over a stretching sheet. The combined effects of non-uniform source/sink and convective boundary condition are taken into consideration. The governing partial differential equations are transformed into ordinary differential equations using similarity variables, which are then solved numerically by using Runge Kutta Fehlberg-45 method with shooting technique. The influence of various parameters on velocity and temperature profiles are illustrated graphically, and discussed in detail. The results indicate that the fluid phase velocity is greater than that of the particle phase for various existing parameters.
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6

Lebelo, Ramoshweu Solomon, K. C. Moloi, C. C. Chitumwa, M. W. R. Sadiki, P. Baloyi, and S. O. Adesanya. "On Thermal Stability Analysis of a Convective and Radiative Slab of Variable Thermal Conductivity with Reactant Consumption." Defect and Diffusion Forum 389 (November 2018): 195–204. http://dx.doi.org/10.4028/www.scientific.net/ddf.389.195.

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Thermal stability in a stockpile of reactive materials that are assumed to lose heat to the surrounding environment by convection and radiation is studied in this article. The reactant (O2) consumption is also considered and the investigation is modeled in a rectangular slab. The complicated combustion process results with nonlinear interactions and therefore, the nonlinear differential equations governing the problem are solved numerically with the Runge-Kutta Fehlberg Method (RKF45) that is coupled with the Shooting Technique. The behaviors of the temperature and the reactant, due to effects of some embedded kinetic parameters, are depicted graphically and discussed accordingly. The results show that kinetic parameters that increase the temperature of the system, correspondingly increase the reactant consumption.
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7

Mehmood, Zaffar, Z. Iqbal, Ehtsham Azhar, and E. N. Maraj. "Nanofluidic Transport over a Curved Surface with Viscous Dissipation and Convective Mass Flux." Zeitschrift für Naturforschung A 72, no. 3 (2017): 223–29. http://dx.doi.org/10.1515/zna-2016-0353.

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AbstractThis article is a numerical investigation of boundary layer flow of nanofluid over a bended stretching surface. The study is carried out by considering convective mass flux condition. Contribution of viscous dissipation is taken into the account along with thermal radiation. Suitable similarity transformations are employed to simplify the system of nonlinear partial differential equations into a system of nonlinear ordinary differential equations. Computational results are extracted by means of a shooting method embedded with a Runge-Kutta Fehlberg technique. Key findings include that velocity is a decreasing function of curvature parameter K. Moreover, Nusselt number decreases with increase in curvature of the stretching surface while skin friction and Sherwood number enhance with increase in K.
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8

Makinde, Oluwole Daniel, and S. R. Mishra. "Chemically Reacting MHD Mixed Convection Variable Viscosity Blasius Flow Embedded in a Porous Medium." Defect and Diffusion Forum 374 (April 2017): 83–91. http://dx.doi.org/10.4028/www.scientific.net/ddf.374.83.

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In this paper, the combined effects of magnetic field, buoyancy forces, nth order chemical reaction, heat source, viscous dissipation, Joule heating and variable viscosity on mixed convection Blasius flow of a conducting fluid over a convectively heated permeable plate embedded in a porous medium is investigated. The fluid properties are assumed to be constant except for the density variation with the temperature and reacting chemical species concentration. The nonlinear governing differential equations were obtained and solved numerically using the Runge-Kutta-Fehlberg method with shooting technique. The dimensionless velocity, temperature and concentration profiles are shown graphically. The effects of pertinent parameters on the skin friction, Nusselt number and Sherwood number are examined. It is found that skin friction decreases while Nusselt number and Sherwood number increase with a decrease in the fluid viscosity in the presence of magnetic field.
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9

Reddy, Konda Jayarami, N. P. Madhusudhana Reddy, Rama Krishna Konijeti, and Dasore Abhishek. "Numerical Investigation of Chemical Reaction and Heat Source on Radiating MHD Stagnation Point Flow of Carreau Nanofluid with Suction/Injection." Defect and Diffusion Forum 388 (October 2018): 171–89. http://dx.doi.org/10.4028/www.scientific.net/ddf.388.171.

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This research work is focused on the effects of combined heat and mass transfer on MHD stagnation point flow of Carreau nanaofluid embedded in porous medium with heat source. Thermal radiation and chemical reaction are also taken into account. The governing non-linear PDEs are transformed into a set of non-linear coupled ODEs which are then solved numerically by using the Runge– Kutta–Fehlberg fourth–fifth order method along shooting technique. The graphical and tabular results elucidate the influence of different non-dimensional governing parameters on the velocity, temperature and concentration fields along with the wall friction, local Nusselt and Sherwood numbers. We found the dual nature of the solutions for suction and injection cases. A good agreement of the present results has been observed by comparing with the existing literature results.
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10

Mondal, Surya Kanta, and Dulal Pal. "Mathematical analysis for Brownian motion of nonlinear thermal bioconvective stagnation point flow in a nanofluid using DTM and RKF method." Journal of Computational Design and Engineering 7, no. 3 (2020): 294–307. http://dx.doi.org/10.1093/jcde/qwaa025.

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Abstract In the present paper, bioconvective stagnation point flow of nanofluid containing gyrotactic microorganisms over a nonlinearly stretching sheet embedded in a porous medium is considered. The scaling group transformation method is introduced to obtain the similarity transformation to convert the governing partial differential equations to a set of ordinary differential equations. The reduced governing nonlinear differential equations are then solved numerically with Runge–Kutta–Fehlberg method. Differential transform method is employed to justify the results obtained by the numerical method. It is found that both the results matched nicely. It is noticed that the density of motile microorganism distribution grows high with an increase in the values of the bioconvection Peclet number. Further, the rate of heat transfer and the rate of mass transfer increase rapidly with an increment in the thermophoresis parameter, heat source parameter, chemical reaction parameter, and Brownian motion parameter, respectively. This work is relevant to engineering and biotechnological applications, such as in the design of bioconjugates and mass transfer enhancement of microfluidics.
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