Nonlinear Two-point Boundary Value Problems Research Articles

On the Solvability of Nonlinear Third-Order Two-Point Boundary Value Problems

Under barrier strips type assumptions we study the existence of C 3 [ 0 , 1 ] —solutions to various two-point boundary value problems for the equation x ‴ = f ( t , x , x ′ , x ″ ) . We give also some results guaranteeing positive or non-negative, monotone, convex or concave solutions.

A new sixth-order approximation for nonlinear two-point boundary value problems: application of single-step alternating group explicit iteration method to engineering problems

We report a new three-point compact sixth-order approximation for the solution of nonlinear two-point boundary value problems $$ {-}w^{\prime\prime} + f\left( {t,w} \right) = 0 $$, subject to natural boundary conditions prescribed at two end points. We also discuss the application of single-step alternating group explicit (SAGE) iteration method to the nonlinear difference equation as a computational tool. The error analysis of the SAGE iteration method is discussed briefly. We have compared the results obtained by using the proposed SAGE iteration method with the results obtained by corresponding two-step alternating group explicit iteration method to demonstrate experimentally the superiority of the proposed method.

Fractal Cubic Spline Methods for Singular Boundary-Value Problems

In this article, we propose a new numerical method based on fractal cubic spline for the numerical approximate solutions of linear and nonlinear singular two-point boundary-value problems. Error analysis of the proposed method is examined and it is shown that proposed method has second-order convergence. Numerical examples are provided to validate the theoretical results.

Magnetohydrodynamic flow of Carreau liquid over a stretchable sheet with a variable thickness

PurposeThe magnetohydrodynamic (MHD) flow problems are important in the field of biomedical applications such as magnetic resonance imaging, inductive heat treatment of tumours, MHD-derived biomedical sensors, micropumps for drug delivery, MHD micromixers, magnetorelaxometry and actuators. Therefore, there is the impact of the magnetic field on the transport of non-Newtonian Carreau fluid in the presence of binary chemical reaction and activation energy over an extendable surface having a variable thickness. The significance of irregular heat source/sink and cross-diffusion effects is also explored.Design/methodology/approachThe leading governing equations are constructed by retaining the effects of binary chemical reaction and activation energy. Suitable similarity transformations are used to transform the governing partial differential equations into ordinary differential equations. Subsequent nonlinear two-point boundary value problem is treated numerically by using the shooting method based on Runge–Kutta–Fehlberg. Graphical results are presented to analyze the behaviour of effective parameters involved in the problem. The numerical values of the mass transfer rate (Sherwood number) and heat transfer rate (Nusselt number) are also calculated. Furthermore, the slope of the linear regression line through the data points is determined in order to quantify the outcome.FindingsIt is established that the external magnetic field restricts the flow strongly and serves as a potential control mechanism. It can be concluded that an applied magnetic field will play a major role in applications like micropumps, actuators and biomedical sensors. The heat transfer rate is enhanced due to Arrhenius activation energy mechanism. The boundary layer thickness is suppressed by strengthening the thickness of the sheet, resulting in higher values of Nusselt and Sherwood numbers.Originality/valueThe effects of magnetic field, binary chemical reaction and activation energy on heat and mass transfer of non-Newtonian Carreau liquid over an extendable surface with variable thickness are investigated for the first time.

A hybrid collocation method for solving highly nonlinear boundary value problems

In this article, a hybrid collocation method for solving highly nonlinear boundary value problems is presented. This hybrid method combines Chebyshev collocation method with Laplace and differential transform methods to obtain approximate solutions of some highly nonlinear two-point boundary value problems of ordinary differential equations. The efficiency of the method is demonstrated by applying it to ordinary differential equations modelling Darcy-Brinkman-Forchheimer momentum problem, laminar viscous flow problem in a semi-porous channel subject to transverse magnetic field, fin problem with a temperature-dependent thermal conductivity, transformed equations modelling two-dimensional viscous flow problem in a rectangular domain bounded by two moving porous walls and two-dimensional constant speed squeezing flow of a viscous fluid between two approaching parallel plates. The results obtained are compared with the existing methods and the results show that the new method is quite reasonable, accurate and efficient.

Nonlinear two-point boundary value problems: applications to a cholera epidemic model

This paper is concerned primarily with constructive mathematical analysis of a general system of nonlinear two-point boundary value problem when an empirically constructed candidate for an approximate solution (quasi-solution) satisfies verifiable conditions. A local analysis in a neighbour- hood of a quasi-solution assures the existence and uniqueness of solutions and, at the same time, provides error bounds for approximate solutions. Applying this method to a cholera epidemic model, we obtain an analytical approximation of the steady-state solution with rigorous error bounds that also displays dependence on a parameter. In connection with this epidemic model, we also analyse the basic reproduction number, an important threshold quantity in the epidemiology context. Through a complex analytic approach, we determine the principal eigenvalue to be real and positive in a range of parameter values.

Decentralized observer-based optimal control using two-point boundary value successive approximation approach application for a multimachine power system

In this paper, we present a new decentralized observer for a class of nonlinear interconnected systems, based on an optimal control law using the two-point boundary value (TPBV) successive approximation approach. This technique, used previously to develop a nonlinear decentralized suboptimal control for a multimachine power system, inspired us to develop a new method to design a decentralized observer-based suboptimal control law for the same system. The TPBV approach is characterized by the transformation of each high order coupling nonlinear TPBV problem into a sequence of linear decoupling TPBV problems that uniformly converge to the optimal control for nonlinear interconnected large-scale systems. Sufficient conditions for the asymptotic stability of the developed feedback control scheme are proposed in terms of linear matrix inequalities (LMIs) constraints, which can be efficiently solved by the LMI optimization techniques, to compute the control and the observation gains of the overall system. We applied this approach to a three-machine power system which generators are nonlinear systems strongly interconnected. We demonstrated clearly, via advanced simulations, that this approach was able to bring good performances, improving effectively transient stability of this power system in few iterative sequences while allowing the reconstruction of its non-measurable state variables.

Solving a class of boundary value problems and fractional Boussinesq-like equation with β-derivatives by fractional-order exponential trial functions

The nonlinear two-point boundary value problem (TPBVP) is converted to a new form of the problem including integral terms. Then the combination of weak-form integral equation method (WFIEM) and special functions create a robust and applicable numerical scheme to solve the problem. To display the accuracy of our method, some examples are investigated. Also, the fractional Boussinesq-like equation involving the β-derivative has been considered that describes the propagation of small amplitude long capillary-gravity waves on the surface of shallow water.

The Modified VIM for Certain Type of the Nonlinear Two-Point Boundary Value Problems

In the current paper, we introduce a different version of the variational iteration method (VIM) that provide an easy and effective procedure for solving nonlinear two-point boundary-value problems appearing in applied and engineering applications. For problems where convergence rate of the original VIM is slow, the method can be readily improved to increase convergence rate. Convergence analysis and error bounds are discussed. Finally, applicability of the proposed method and accuracy are examined by solving three problems.

A Unified Approach for Solving Linear and Nonlinear Odd-Order Two-Point Boundary Value Problems

In this article, we propose new spectral solutions for odd-order two-point boundary value problems. A numerical algorithm based on the use of collocation methods is implemented and analyzed. A new matrix of derivatives of certain Legendre basis polynomials is derived to serve in deriving the proposed algorithm. Our algorithm has the advantage that it is applicable to both linear and nonlinear odd-order two-point boundary value problems. The convergence and error analysis of the suggested method are investigated. Five illustrative examples supported with comparisons are displayed to ensure the efficiency and applicability of the proposed algorithm.

An adaptive local discontinuous Galerkin method for nonlinear two-point boundary-value problems

In this paper, we propose an adaptive mesh refinement (AMR) strategy based on a posteriori error estimates for the local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the form $u^{\prime \prime }=f(x,u),\ x\in [a,b]$ subject to some suitable boundary conditions at the endpoint of the interval [a, b]. We first use the superconvergence results proved in the first part of this paper as reported by Baccouch (Numer. Algorithm. 79(3), 697–718 2018) to show that the significant parts of the local discretization errors are proportional to (p + 1)-degree Radau polynomials, when polynomials of total degree not exceeding p are used. These new results allow us to construct a residual-based a posteriori error estimators which are obtained by solving a local residual problem with no boundary conditions on each element. The proposed error estimates are efficient, reliable, and asymptotically exact. We prove that, for smooth solutions, the proposed a posteriori error estimates converge to the exact errors in the L2-norm with order of convergence p + 3/2. Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p ≥ 1. Several numerical results are presented to validate the theoretical results and to show the efficiency of the grid refinement strategy.

A fixed point iteration method using Green's functions for the solution of nonlinear boundary value problems over semi-Infinite intervals

ABSTRACTIn this paper, an iterative method is introduced for the numerical solution of a class of nonlinear two-point boundary value problems (BVPs) on semi-infinite intervals. The underlying strategy behind this novel approach is to construct a tailored integral operator that is expressed in terms of a Green's function for the corresponding linear differential operator of the BVP. Then, two well-known fixed point iterations, including Picard's and Krasnoselskii-Mann's schemes, are applied to this integral operator that results in this new iterative technique. A proof of convergence of the numerical scheme, based on the contraction principle, is included. We demonstrate the reliability, fast convergence, applicability of the method and compare its performance, using some relevant test examples that appear in the literature.

Circular flow around a turning point in an annular area between two coaxial porous cylinders

In order to investigate a suction-driven circular flow within an annular tube formed by two coaxial porous cylinders, a mathematical model expressed as a nonlinear two-point boundary-value problem is achieved. In seeking solutions of the problem, the results obtained reveal the existence of a hydrodynamic turning point among the main physical settings of the study. The dynamics of the fluid is examined through three solution branches Fr, S1 and S2 that form this turning point. By increasing the Reynolds number above the turning point, regions where the fluid moves in the counterclockwise direction are observed through the first branch Fr and the secondary branch of type S2 near the inner and the outer cylinders, respectively. Due to this described behavior, solutions of types Fr and S2 seem to be mirror images of each other. A flow of the boundary layer type takes place through the secondary branch of type S1 for great values of the Reynolds number. This boundary layer causes the radial velocity to approach a linear profile at the same constant curve for different Reynolds numbers.

Replacing the finite difference methods for nonlinear two-point boundary value problems by successive application of the linear shooting method

This paper studies numerical solution of nonlinear two-point boundary value problems for second order ordinary differential equations. First, it establishes a connection between the finite difference method and the quasi-linearization method. We prove that using finite differences to discretize the sequence of linear differential equations arising from quasi-linearization (Newton method on operator level) leads to the usual iteration formula of the Newton finite difference method. From the provided derivation, it can easily be inferred that such a relation holds also for the Picard and the constant-slope methods. Based on this result, we propose a way of replacing the Newton, Picard, and constant-slope finite difference methods by respective successive application of the linear shooting method. This approach has a number of advantages. It removes the necessity of solving systems of algebraic equations, hence working with matrices, altogether. Compared to the usual finite difference method with general solver, it reduces the number of computational operations from O ( N 3 ) , where N is the number of mesh-points, to only O ( N ) .

Dead-Core Solutions to Simple Catalytic Reaction Problems in Chemical Engineering

The catalytic chemical reaction is usually carried out in a pellet where the catalyst is distributed throughout its porous structure. The selectivity, yield and productivity of the catalytic reactor often depend on the rates of chemical reactions and the rates of diffusion of species involved in the reactions in the pellet porous space. In such systems, the fast reaction can lead to the consumption of reactants close to the external pellet surface and creation of the dead core where no reaction occurs. This will result in an inefficient use of expensive catalyst. In the discussed simplified diffusion-reaction problems a nonlinear reaction term is of power-law type with a small positive reaction exponent. Such reaction term represents the kinetics of catalytic reaction accompanied by a strong adsorption of the reactant. The ways to calculate the exact solutions possessing dead cores are presented. It was also proved analytically that the exact solution of the nonlinear two-point boundary value problem satisfies physical a-priori bounds. Furthermore, the approximate solutions were obtained using the orthogonal collocation method for pellets of planar, spherical and cylindrical geometries. Numerical results confirmed that the length of the dead core increases for the more active catalysts due to the larger values of the reaction rate constant. The dead core length also depends on the pellet geometry.

A continuous solution of solving a class of nonlineartwo point boundary value problem using Adomian decomposition method

Abstract In this paper, a modified version of Adomian decomposition method is used to solve a class of nonlinear boundary value problems. It is easy to solve IVP using ADM but there exist some difficulties in solving BVP. But by using our modification of ADM, we solve the BVP by easily and directly method. As well we get a continuous solution of BVP. This technique gives a continuous solution and improves the work done by El-Kalla 2013. Some numerical examples are introduced to verify the efficiency of the new results.

Numerical techniques for behavior of incompressible flow in steady two-dimensional motion due to a linearly stretching of porous sheet based on radial basis functions

AbstractIn this paper the boundary layer flow of a micro-polar fluid due to a linearly stretching sheet which is a non-linear system two-point boundary value problem (BVP) on semi-infinite interval has been considered. This the sheets are included the suction and injection. We solve this problem by two different collecation approaches and compare their results with solution of other methods. The proposed approaches are equipped by the direct (DRBF) and indirect radial basis functions (IRBF). Direct approach (DRBF) is based on a differential process and indirect approach (IRBF) is based on an integration process. These methods reduce solution of the problem to solution of a system of algebraic equations. Numerical results and residual norm show that the IRBF performs better than the common DRBF, and has an acceptable accuracy and high rate of convergence of IRBF process.

A compact finite difference method for a general class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions

In this paper, we develop and analyze a high order compact finite difference method (CFDM) for solving a general class of two-point nonlinear singular boundary value problems with Neumann and Robin boundary conditions arising in various physical models. Convergence result of this method is established through matrix analysis approach. To illustrate the applicability and accuracy of the method, we consider nine numerical examples, including heat conduction in the human head, equilibrium of isothermal gas sphere, oxygen-diffusion in a spherical cell and reaction–diffusion process in a spherical permeable catalyst. It is shown that the computational order of convergence of the proposed CFDM is four. The obtained results are compared with those obtained by other existing numerical methods.

Study of a nonlinear boundary-value problem of geophysical relevance

We present some results on the existence and uniqueness of solutions of a two-point nonlinear boundary value problem that arises in the modeling of the flow of the Antarctic Circumpolar Current.

Application of Optimal Homotopy Asymptotic Method to Some Well-Known Linear and Nonlinear Two-Point Boundary Value Problems

The objective of this paper is to obtain an approximate solution for some well-known linear and nonlinear two-point boundary value problems. For this purpose, a semianalytical method known as optimal homotopy asymptotic method (OHAM) is used. Moreover, optimal homotopy asymptotic method does not involve any discretization, linearization, or small perturbations and that is why it reduces the computations a lot. OHAM results show the effectiveness and reliability of OHAM for application to two-point boundary value problems. The obtained results are compared to the exact solutions and homotopy perturbation method (HPM).