The purpose of the present paper is to introduce a method, probably for the first time, to predict the multiplicity of the solutions of nonlinear boundary value problems. This procedure can be easily ...applied on nonlinear ordinary differential equations with boundary conditions. This method, as will be seen, besides anticipating of multiplicity of the solutions of the nonlinear differential equations, calculates effectively the all branches of the solutions (on the condition that, there exist such solutions for the problem) analytically at the same time. In this manner, for practical use in science and engineering, this method might give new unfamiliar class of solutions which is of fundamental interest and furthermore, the proposed approach convinces to apply it on nonlinear equations by today’s powerful software programs so that it does not need tedious stages of evaluation and can be used without studying the whole theory. In fact, this technique has new point of view to well-known powerful analytical method for nonlinear differential equations namely homotopy analysis method (HAM). Everyone familiar to HAM knows that the convergence-controller parameter plays important role to guarantee the convergence of the solutions of nonlinear differential equations. It is shown that the convergence-controller parameter plays a fundamental role in the prediction of multiplicity of solutions and all branches of solutions are obtained simultaneously by one initial approximation guess, one auxiliary linear operator and one auxiliary function. The validity and reliability of the method is tested by its application to some nonlinear exactly solvable differential equations which is practical in science and engineering.
The purpose of this paper is to present a kind of analytical method so-called
Predictor homotopy analysis method (PHAM) to predict the multiplicity of the solutions of nonlinear differential ...equations with boundary conditions. This method is very useful especially for those boundary value problems which admit multiple solutions and furthermore is capable to calculate all branches of the solutions simultaneously. As illustrative examples, the method is checked by the model of mixed convection flows in a vertical channel and a nonlinear model arising in heat transfer which both admit multiple (dual) solutions.
The new rational
a
-polynomials are used to solve the Falkner-Skan equation. These polynomials are equipped with an auxiliary parameter. The approximated solution to the Falkner-Skan equation is ...obtained by the new rational
a
-polynomials with unknown coefficients. To find the unknown coefficients and the auxiliary parameter contained in the polynomials, the collocation method with Chebyshev-Gauss points is used. The numerical examples show the efficiency of this method.
The problem of steady, laminar, mixed convection boundary-layer flow over a vertical cone embedded in a porous medium saturated with a nanofluid is studied, in the presence of thermal radiation. The ...model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis with Rosseland diffusion approximation. The cone surface is maintained at a constant temperature and a constant nanoparticle volume fraction. The resulting governing equations are non-dimensionalized and transformed into a non-similar form and then solved by Keller box method. A comparison is made with the available results in the literature, and our results are in very good agreement with the known results. A parametric study of the physical parameters is made and a representative set of numerical results for the local Nusselt and Sherwood numbers are presented graphically. Also, the salient features of the results are analyzed and discussed.
A numerical scheme for the Brinkman–Forchheimer momentum equation modeling flow in a saturated porous duct is considered. There is no natural time variable we introduce a fictitious time variable, ...and, upon discretization of the two spatial variables, we obtain a system of ordinary differential equations in the fictitious time variable. The resulting system of ordinary differential equations is solved via a geometric numerical integration method known as the group preserving scheme. The group preserving scheme ensures the preservation of the group and cone structure of a system, resulting in a solution with the same asymptotic behavior as the original continuous system, avoiding spurious solutions or ghost fixed points. This fictitious time integration method allows us to obtain numerical solutions with low residual errors, and we compare our results favorably against analytical and numerical results present in the literature. Stability and convergence analysis of the method have been performed. Using these numerical solutions, we are able to discuss the effects of the various physical parameters, such as the inertial coefficient, viscosity of the fluid, effective viscosity, permeability of the porous media, and adverse applied pressure gradient on the fluid velocity through a porous duct modeled under the Brinkman–Forchheimer momentum equation.
This paper aims to present complete analytic solution to heat transfer of a micropolar fluid through a porous medium with radiation. Homotopy analysis method (HAM) has been used to get accurate and ...complete analytic solution. The analytic solutions of the system of nonlinear ordinary differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. The velocity and temperature profiles are shown and the influence of coupling constant, permeability parameter and the radiation parameter on the heat transfer is discussed in detail. The validity of our solutions is verified by the numerical results (fourth-order Runge–Kutta method and shooting method).
► Unsteady boundary layer flow of a special third grade fluid are considered by HAM. ► The boundary layer thickness is an increasing function of third grade parameter. ► Boundary layer thickness in ...injection is larger than suction. ► HAM applied for large values of a non-Newtonian parameter in injection/suction cases.
Analytic solution for the time-dependent boundary layer flow over a moving porous surface is derived by using homotopy analysis method (HAM). A special third grade fluid model has been used in the problem formulation. The obtained HAM solution is also compared with the numerical solution and a reasonable agreement is noted.
This paper investigates two basic steps of the homotopy analysis method (HAM) when applied to nonlinear boundary value problems of the chemical reaction kinetics, namely (1) the prediction and (2) ...the effective calculation of multiple solutions. To be specific, the approach is applied to the dual solutions of an exactly solvable reaction-diffusion model for porous catalysts with apparent reaction order n=-1. It is shown that (i) the auxiliary parameter ℏ which controls the convergence of the HAM solutions in general plays a basic role also in the prediction of dual solutions, and (ii) the dual solutions can be calculated by starting the HAM-algorithm with one and the same initial guess. It is conjectured that the features (1) and (2) hold generally in use of HAM to identify and to determine the multiple solutions of nonlinear boundary value problems.
In this paper, a new method is used based on polynomials equipped with a parameter to solve two parabolic inverse problems. These inverse problems have nonlocal boundary conditions and ...over-determination of data that make it difficult to solve these problems. In this method, we use the combination of the finite difference method and the finite element method. In each point
, a nonlinear equation system is solved via the least-squares method, and then, we obtain an approximate function for the solution of the problem by using the interpolation of these points.
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