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We propose an adaptive and explicit Runge–Kutta–Fehlberg method coupled with a fourth-order compact scheme to solve the American put options problem. First, the free boundary problem is converted into a system of partial differential equations with a fixed domain by using logarithm transformation and taking additional derivatives. With the addition of an intermediate function with a fixed free boundary, a quadratic formula is derived to compute the velocity of the optimal exercise boundary analytically. Furthermore, we implement an extrapolation method to ensure that at least, a third-order accuracy in space is maintained at the boundary point when computing the optimal exercise boundary from its derivative. As such, it enables us to employ fourth-order spatial and temporal discretization with Dirichlet boundary conditions for obtaining the numerical solution of the asset option, option Greeks, and the optimal exercise boundary. The advantage of the Runge–Kutta–Fehlberg method is based on error control and the adjustment of the time step to maintain the error at a certain threshold. By comparing with some existing methods in the numerical experiment, it shows that the present method has a better performance in terms of computational speed and provides a more accurate solution.

The intention of the present work is to study the stability analysis of heat transfer enhancement occurring due to the influence of significant properties variation of fluids in the presence of thermal radiation with an aid of suspended hybrid nanofluids. The mathematical equations are converted into a pair of self-similarity equations by applying appropriate transformation. Runge Kutta Fehlberg 45th order method is applied to solve the reduced similarity equivalences numerically. The flow and energy transfer characteristics are studied for distinct values of important factors to obtain better perception of the problem. According to graphical results, heat transfer enhancement is higher for larger values of radiation parameter (R) and higher values of Prandtl number resulted in heat transfer reduction.

Here formulation and computations are presented to introduce the novel concept of activation energy in chemically reacting stagnation point flow towards a stretching sheet. Constitutive expression for Cross liquid is taken into account. Magnetic field is utilised in the transverse direction. Application of suitable variables generates the non-linear differential systems. Numerical solution by Runge–Kutta–Fehlberg approach is presented. Characteristics for the significant variables like Weissenberg number, Hartmann number, Schmidt number, activation energy chemical reaction parameter, velocity ratio parameter and Prandtl number on the physical quantities are addressed through graphs and tables. Our computations reveal that species concentration rises via larger activation energy parameter whereas it decays when Schmidt number is incremented. The Weissenberg number has opposite characteristics for local Nusselt and Sherwood numbers when compared with surface drag force.

Purpose – The purpose of this paper is to examine the Sisko fluid model over a stretching cylinder with heat transfer and magnetohydrodynamics. Design/methodology/approach – The boundary layer approach is employed to simplify the governing equations. Suitable similarity transformations are used to transform the governing partial differential equations into ordinary differential equations. In order to solve this system of ordinary differential equations numerically, shooting method in conjunction with Runge-Kutta-Fehlberg method is used. Findings – The effects of physical parameters involved in velocity and temperature profiles are shown through graphs. It is observed that Sisko fluid parameter and curvature parameter enhances fluid velocity while motion of fluid is retarded by increasing magnetic field strength. Additionally temperature of fluid raise with curvature parameter while it fall down for larger values of Prandtl number. Skin friction coefficient and Nusselt number are computed and presented in graphs and tables for further analysis. It can be seen that curvature parameter increases both skin friction and Nusselt number while magnetic field and Prandtl number decayed skin friction and Nusselt number, respectively. Also Sisko parameter enlarges skin friction coefficient. The accuracy of solution is verified by comparing it with existing literature. Originality/value – The computed results are interested for industrial and engineering processes, especially in cooling of nuclear reactors.

On the basis of the error correction method developed recently, an algorithm, so-called error embedded error correction method, is proposed for initial value problems. Two deferred equations are used to approximate the solution and the error, respectively, at each integration step. For the solution, the deferred equation, which is based on a modified Euler’s polygon including the information of both the solution and its estimated error at the previous integration step, is solved with the classical fourth-order Runge–Kutta method. For the error, the deferred equation, which is based on a local Hermite cubic polynomial with three pieces of information—the solution, its estimated error at the previous step, and the constructed solution—is solved by the seventh-order Runge–Kutta–Fehlberg method. The constructed algorithm controls the error and possesses a good behavior of error bound in a long time simulation. Numerical experiments are presented to validate the proposed algorithm.

A methodology that can generate the optimal coefficients of a numerical method with the use of an artificial neural network is presented in this work. The network can be designed to produce a finite difference algorithm that solves a specific system of ordinary differential equations numerically. The case we are examining here concerns an explicit two-stage Runge–Kutta method for the numerical solution of the two-body problem. Following the implementation of the network, the latter is trained to obtain the optimal values for the coefficients of the Runge–Kutta method. The comparison of the new method to others that are well known in the literature proves its efficiency and demonstrates the capability of the network to provide efficient algorithms for specific problems.

This paper presents the study of convective heat transfer characteristics of an incompressible dusty fluid past a vertical stretching sheet. The governing partial differential equations are reduced to nonlinear ordinary differential equations by using similarity transformation. The transformed equations are solved numerically by applying Runge Kutta Fehlberg fourth-fifth order method (RKF45 Method). Here obtained non-dimensional velocity and temperature profiles has been carried out to study the effect of different physical parameters such as fluid-particle interaction parameter, Grashof number, Prandtl number, Eckert number. Comparison of the obtained numerical results is made with previously published results.

Taylor-series integration is a numerical integration technique that computes the Taylor series of state variables using recurrence relations and uses this series to propagate the state in time. A Taylor-series integration reentry integrator is developed and compared with the fifth-order Runge–Kutta–Fehlberg integrator to determine whether Taylor-series integration is faster than traditional integration methods for reentry applications. By comparing the central processing unit times of the integrators, Taylor-series integration is indeed found to be faster for integration without wind and slower with wind, unless the error tolerance is 10−8 or lower. Furthermore, it is found that reducing step sizes to prevent integration over discontinuities is not only needed for Taylor-series integration to obtain maximum accuracy but also for Runge–Kutta–Fehlberg methods. In that case, the Runge–Kutta–Fehlberg integrator does become several times slower than Taylor-series integration.

A new algorithm for the integration of the intrinsic reaction coordinate path on a potential energy surface is proposed, based in solving the set of differential equations associated to the corresponding characteristic curve, rather than the differential equation of the curve itself. The integration of this set of differential equations is carried out by employing a Runge–Kutta–Fehlberg with τ-stage and p-algebraic order technique. In addition, an update Hessian matrix formula is reported for this specific algorithm. The examples show that the proposed algorithm is numerically stable, with no extra computational effort with respect to the standard algorithms.

An explicit time-stepping method is developed for adaptive solution of time-dependent partial differential equations with first order derivatives. The space is partitioned into blocks and the grid is refined and coarsened in these blocks. The equations are integrated in time by a Runge---Kutta---Fehlberg (RKF) method. The local errors in space and time are estimated and the time and space steps are determined by these estimates. The method is shown to be stable if one-sided space discretizations are used. Examples such as the wave equation, Burgers' equation, and the Euler equations in one space dimension with discontinuous solutions illustrate the method.

The present work investigates an appropriate way to calculate the variations of nuclides composition in the reactor core during operations. Specific Software has been designed for this purpose using C#. The mathematical approach is based on the solution of Bateman differential equations using a Runge–Kutta–Fehlberg method. Material depletion at constant flux and constant power can be calculated with this software. The inputs include reactor power, time step, initial and final times, order of Taylor Series to calculate time dependent flux, time unit, core material composition at initial condition (consists of light and heavy radioactive materials), acceptable error criterion, decay constants library, cross sections database and calculation type (constant flux or constant power). The atomic density of light and heavy fission products during reactor operation is obtained with high accuracy as the program outputs. The results from this method compared with analytical solution show good agreements.

A novel method with automatic step-size adjustment for propagation equation calculations in fiber Raman amplifiers is proposed, for the first time to authors' knowledge, in this paper. An effective shooting algorithm for the two-point boundary value problem is also constructed. Numerical results demonstrate that the computing speed of the proposed method is increased more than four times by comparison with the classical Runge–Kutta–Fehlberg method, the step-size is extended greatly, and the stability is improved. Simulation results also show that the new shooting algorithm can be used to solve Raman amplifier propagation equations on various conditions including co-, counter- and bi-directional pump schemes.

In this paper, an analysis has been carried out to investigate the hydromagnetic fluid flow of dusty fluidwith thermal radiation at vertical stretching sheet. The behavior of velocity and temperature profile ofhydromagnetic fluid flow with fluid particle suspension is analyzed by using Runge Kutta Fehlberg forthfifthorder method (RKF45 Method). These solutions are presented and discussed for different parametersof interest such as fluid particle interaction parameter, the magnetic parameter, the radiation parameter,Grashof number, Eckert number and Prandtl number on the flow.

This paper proposes a method for the semi-analytical solution of the coupled linear differential equations that describe the linear modal coupling arising in few mode fibers (FMFs) due to waveguide imperfections. The semi-analytical solutions obtained can be used for the modification of the split step Fourier method to include the respective linear effects. Considering a fiber with six linearly polarized modes (or ten modes, taking into account the degenerate versions) the solutions obtained proved to be accurate when compared to the numerical solutions obtained through the Runge-Kutta-Fehlberg method.

There is no an explicit solution for is proposed problem in 1963 by Lorentz. Generally, this system of differentials equations is solved by the method oftrapezoidal integration with extrapolation of Richardson or by the improved method of Euler. In this work an altemative solution by the method of Runge Kutta Fehlberg ís presented, which requires only six evaluat.ions ofthe function, thus reducing the time ofcalculation., No hay solución explicita para el problema planteado en 1963 por Lorentz, por lo general se resuelve este sistema de ecuaciones diferenciales por el método de integración trapezoidal con extrapolación de Richarson o por el método mejorado de Euler. En este trabajo se plantea una solución alternativa por el método de Runge Kutta Fehlberg que requiere solo seis evaluaciones de la función, reduciendo el tiempo de cálculo.

In this paper a new fourth and fifth-order numerical solution of linear Volterra integro-differential equation is discussed. One popular technique that uses here for error control is called the Runge-Kutta-Fehlberg method for Ordinary Differential Equation (ODE) part and Newton-Cotes formulae for integral parts.

The present problem is the steady boundary layer flow and heat transfer of a hyperbolic tangent fluid flowing over a vertical exponentially stretching cylinder in its axial direction. After applying usual boundary layer with a suitable similarity transformation to the given partial differential equations and the boundary conditions, a system of coupled nonlinear ordinary differential equations is obtained. This system of ordinary differential equations subject to the boundary conditions is solved with the help of Runge–Kutta–Fehlberg method. The effects of the involved parameters such as Reynolds numbers, Prandtl numbers, Weissenberg numbers and the natural convection parameter are presented through the graphs. The associated physical properties on the flow and heat transfer characteristics that is the skin friction coefficient and Nusselt numbers are presented for different parameters.