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The purpose of this paper is to demonstrate an effective and robust numerical solution for Raman fiber amplifier (RFA) equations which have no explicit solution. MATLAB BVP solvers are addressed for the solution.

The continuation method proposed for the solution of RFA equations using MATLAB BVP solvers is explained. Scripts for improving the power values at the boundaries with continuation, extending fiber length with continuation and calculation of the analytical partial derivatives using the MATLAB Symbolic toolbox are introduced. Comparisons among the different MATLAB BVP solvers have been made. Using the continuation method, signal evolutions for different kinds of RFA amplifier configurations are plotted.

The paper finds that MATLAB BVP solver with the continuation method can be used in the design of various kinds of RFAs for high powers/long gain fiber spans.

The paper will assist the fiber optic research community who suffer from two or more point boundary‐value problems. Moreover, the stiffness of the signal evolution which is faced with high pump powers and/or long fiber lengths can be solved with continuation. This superiority of the solver can be used to overcome any stiff changes of the signals for future studies.

The increased research interests and practical demands for RFAs have been calling for reasonable and efficient means for the performance evaluation of RFAs before the real amplifiers are fabricated. The solution method presented in this paper will be an efficient means for the solution of this issue.

MATLAB BVP solvers have been proven to be effective for the numerical solution of RFAs with multiple pumps and signal waves. Using the continuation method, in a distributed RFA with ten pump sources, 2,400 mW total input pump power is achieved. The improvement of the total power is about 1.4 times compared with those of the previously reported methods. Using the MATLAB BVP solvers, total power/fiber span can be improved further using the continuation process with the cost of computational time. This is a notable and promising improvement from a RFA designer's point of view.

The aim of the paper is to demonstrate a fast numerical solution for Raman fiber amplifier equations using proposed guess functions and MATLAB intrinsic properties. MATLAB BVP solvers are addressed for the solution.

The guess functions proposed for the solution of RFA equations using MATLAB BVP solvers are derived from Taylor expansion of pump and signal wave near the boundary to specifically obtain convergence for the initial mesh point. The guess functions increase simulation speed significantly. In order to improve the simulation speed further, vectorization and analytical Jacobians are introduced. Comparisons among bvp4c and bvp5c have been made with respect to total pump power, number of signals, vectorization with/without analytical Jacobians, fiber length, relative tolerance and continuation method. The simulations are performed to determine the effect of the run time on the choice of the number of equally spaced mesh points (N) in the initial guess, and thus optimal N values are found.

MATLAB BVP solvers have been proven to be effective for the numerical solution of RFAs with the proposed guess functions. In particular, with vectorizing, run time reduction is between 2.1 and 5.4 times for bvp4c and between 1.6 and 2.1 times for bvp5c and in addition to vectorizing, with the introduction of the analytical Jacobians, the reduction is between 2.4 and 6.2 times for bvp4c and 1.7 and 2.2 times for bvp5c, respectively, depending on the total pump power between 1,000 mW and 2,000 mW and the number of signals. Also, simulation results show that the efficiency of the solution with proposed guess functions is improved more than six times compared with those of previously reported continuation methods. Results show that the proposed guess functions with the vectorization and analytical Jacobians can be used for the performance evaluation of RFAs for the high power systems/long gain fiber span.

The robust improvement of the solution proposed in this paper lies in the fact that the derived guess functions for the RFAs are highly effective in the sense that they assist the solver to converge to the solution for any total pump power value in a wide range from 1 to 3,000 mW and for any fiber lengths ranging 1 to 200 km which are used in practical applications. Hence, it is practicable for the performance evaluation of the existing RFA networks.

The novelty of this method is that, starting with the co‐propagating single pump and signal RFA schema, the authors derived the guess function specifically for the initial mesh points rather than using its analytical approximations. Moreover, the solution is generalized for co‐/counter propagating pumps/signals with the curve fitted coefficient(s).

The purpose of this paper is to demonstrate an effective and faster numerical solution for nonlinear‐coupled differential equations describing fiber amplifiers which have no explicit solution. MATLAB boundary value problem (BVP) solver of bvp6c function is addressed for the solution.

Coding method with the bvp6c is introduced, signal evolution, threshold calculation method is introduced, gain and noise figure are plotted and superiority of the bvp6c solver is compared with the Newton‐Raphson method.

bvp6c function appears to be an effective tool for the solution fiber amplifier equations and can be used for different pump configurations of BFAs and RFAs. The excellent agreement between the proposed and reported results shows the reliability of the proposed threshold power calculation method.

The paper eases the work of the fiber optic research community, who suffer from two point BVPs. Moreover, the stiffness of the signal evolution which is faced with high pump powers and/or long fiber lengths can be solved with continuation. This superiority of the solver can be used to overcome any stiff changes of the signals for the future studies.

The main outcome of this paper is the numerically calculation of the threshold values of fiber amplifiers without the necessity of the experiment. The robustness improvement of the solution is that the solver is able to solve the equations even with the poor guess values and the solution can be obtained without the necessity of analytical Jacobian matrix.

MATLAB bvp6c solver has proven to be effective for the numerical solution of nonlinear‐coupled intensity differential equations describing fiber amplifiers with two‐point boundary values. Beside the signal evolution, thresholds of Brillouin and Raman fiber amplifiers can also be calculated by using the proposed solver. This is a notable and promising improvement of the paper, at least from a fiber optic amplifier designer point of view.

The purpose of this paper is to report the effect of radiation on flow of a magneto‐micropolar fluid past a continuously moving plate with suction and blowing.

The governing equations are transformed into dimensionless nonlinear ordinary differential equations by similarity transformation. Analytical technique, namely the homotopy perturbation method (HPM) combining with Padé approximants and finite difference method, are used to solve dimensionless non‐linear ordinary differential equations. The skin friction coefficient and local Nusselt numbers are also calculated. Beside this, the comparison of the analytical solution with numerical solution is illustrated by the graphs for different values of dimensionless pertinent parameters.

The authors have studied laminar magneto‐micropolar flow in the presence of radiation by using HPM‐Padé and finite difference methods. Results obtained by HPM‐Padé are in excellent agreement with the results of numerical solution.

The HPM‐Padé is used in a direct way without using linearization, discritization or restrictive assumption. The authors have attempted to show the capabilities and wide‐range applications of the HPM‐Padé in comparison with the finite difference solution of magneto‐micropolar flow in the presence of radiation problem.

This paper aims to report theoretical and numerical results for the problem of laminar axisymmetric flow of hybrid nanofluid over a permeable non-linearly stretching/shrinking sheet with radiation effect.

The numerical solutions of the arising boundary value problem are obtained using the function bvp4c from MATLAB for different values of the governing parameters.

It is found that the solutions of the ordinary (similarity) differential equations have two branches, upper and lower branch solutions, in a certain range of the stretching/shrinking and suction parameters. To establish which of these solutions are stable and which are not, a stability analysis has been performed.

To the best of the authors’ knowledge, present results are original and new for the study of fluid flow and heat transfer over a stretching/shrinking surface, as they successfully extend the problem considered by Mustafa et al. (2015) to the case of hybrid nanofluids.

The purpose of this paper is to scrutinize the effects of nonlinear thermal radiation and thermal stratification effects on the flow of three-dimensional Eyring-Powell 36 nm alumina-water nanofluid within the thin boundary layer in the presence of quartic autocatalytic kind of chemical reaction effects, and to unravel the effects of a magnetic field parameter, random motion of the tiny nanoparticles and volume fraction on the flow.

The chemical reaction between homogeneous (Eyring-Powell 36 nm alumina-water) bulk fluid and heterogeneous (three molecules of the catalyst at the surface) in the flow of magnetohydrodynamic three-dimensional flow is modeled as a quartic autocatalytic kind of chemical reaction. The electromagnetic radiation which occurs within the boundary layer is treated as the nonlinear form due to the fact that Taylor series expansion may not give full details of such effects within the boundary layer. With the aid of appropriate similarity variables, the nonlinear coupled system of partial differential equation which models the flow was reduced to ordinary differential equation boundary value problem.

A favorable agreement of the present results is obtained by comparing it for a limiting case with the published results; hence, reliable results are presented. The concentration of homogeneous bulk fluid (Eyring-Powell nanofluid) increases and decreases with ϕ and P_r, respectively. The increase in the value of magnetic field parameter causes vertical and horizontal velocities of the flow within the boundary layer to decrease significantly. The decrease in the vertical and horizontal velocities of Eyring-Powell nanofluid flow within the boundary layer is guaranteed due to an increase in the value of M. Concentration of homogeneous fluid increases, while the concentration of the heterogeneous catalyst at the wall decreases with M.

Considering the industrial applications of thermal stratification in solar engineering and polymer processing where the behavior of the flow possesses attributes of Eyring-Powell 36 nm alumina-water, this paper presents the solution of the flow problem considering 36 nm alumina nanoparticles, thermophoresis, stratification of thermal energy, Brownian motion and nonlinear thermal radiation. In addition, the aim and objectives of this paper fill such vacuum in the industry.

The purpose of this paper is to provide an insight into the influence of gyrotactic microorganisms and Hall effect on the boundary layer flow of 29 nm CuO-water mixture on the upper pointed surface of a rocket, over the bonnet of a car and upper pointed surface of an aircraft. This is true since all these objects are examples of an object with variable thickness.

The simplification of Rosseland approximation (Taylor series expansion of T⁴ about T_∞) is avoided; thus, two different parameters relating to the study of nonlinear thermal radiation are obtained. The governing equation is non-dimensionalized, parameterized and solved numerically.

Maximum vertical and horizontal velocities of the 29 nm CuO-water nanofluid flow is guaranteed at a small value of Peclet number and large value of buoyancy parameter depending on the temperature difference. When the magnitude of thickness parameter χ is small, cross-flow velocity decreases with the velocity index and the opposite effect is observed when the magnitude of χ is large.

Directly or indirectly, the importance of the fluid flow which contains 29 nm CuO nanoparticle, water, and gyrotactic microorganisms in the presence of Hall current has been pointed out as an open question in the literature due to its relevance in imaging, ophthalmological and translational medicine informatics.

The aim of the current study is proposing a novel framework to attain the optimum value of a flexible arm manipulator parameters for payload launching missions.

The proposed scheme is based on optimal control approach and combines direct and indirect search methods while considering the actuator capacity.

Three nonlinear parameter-optimization problems will be solved to illustrate how the proposed algorithm can be exploited. Employing variational based nonlinear optimal control within the suggested framework, the answer of these problems is highly intertwined to the solution of a set of differential equations with split boundary values. To solve the obtained boundary value problem (BVP), the related solver of MATLAB® software, bvp6c, will be employed. The achieved simulation results support the worth of the developed procedure.

For the first time, the optimal parameters of a flexible link robot for object launching are found in the current research. In addition, the actuator saturation limits are considered which enhances the applicability of the suggested method in the real world applications.

As transferring biological fluid through an artery is nowadays a pivotal subject, the purpose of this paper is to study the mathematical model of hybrid nanofluid flow comprising pure blood as base fluid and TiO₂ and Ag as nanoparticles through the porous channel, which can be an applicable model for drug delivery.

Both walls of the channel have different permeability, which enables the fluid to enter and exit, and variable height, which dilates and squeezes at the uniform rate. By taking advantage of the similarity transformation technique, governing equations have been converted into a system of the non-linear ordinary differential equation. This problem is solved numerically by utilizing BVP4C built-in function in MATLAB software to explore the impacts of pertinent parameters.

The plots of velocity and temperature profile, normal pressure distribution and wall shear stress, as well as Nusselt number for involved parameters, are presented and the logic and physical reasons beyond them are highlighted. It has been observed that the asymmetry of the channel, caused by different permeability at walls, affects the nature of flow significantly.

To the best of the authors’ knowledge, no one has ever attempted to study the flow through a deformable porous channel with blood as a base fluid and as hybrid nanoparticles to describe medical phenomena and treatment applications. Indeed, the achievements of this paper are purely original and the numerical results were never published by any researcher.

The current investigation is communicated to analyze the characteristics of squeezed second grade nanofluid flow enclosed by infinite channel in the existence of both heat generation and variable viscosity. The leading non-linear energy and momentum PDEs are converted into non-linear ODEs by using suitable analogous approach.

Then the acquired non-linear problem is numerically calculated by using Bvp4c (built in) technique in MATLAB.

The influence of certain appropriate physical parameters, namely, squeezed number, fluid parameter, Brownian motion, heat generation, thermophoresis parameter, Prandtl number, Schmidt number and variable viscosity parameter on temperature, velocity and concentration distributions are studied and deliberated in detail. Numerical calculations of Sherwood number, Nusselt number and skin friction for distinct estimations of appearing parameters are analyzed through graphs and tables. It is examined that for large values of squeezing parameter, the velocity profile increases, whereas opposite behavior is noticed for large values of variable viscosity and fluid parameter. Moreover, temperature profile increases for large values of Brownian motion, thermophoresis parameter and squeezed parameter and decreases by increases Prandtl number and heat generation. Moreover, concentration profile increases for large values of Brownian motion parameter and decreases by increases thermophoresis parameter, squeezed parameter and Schmidt number.

No one has ever taken infinite squeezed channel having second grade fluid model with variable viscosity and heat generation.