Nonlinear Boundary Value Problems Research Articles

Nonlinear elliptic boundary value problems with convection term and Hardy potential

Abstract In this paper, we deal with a nonlinear elliptic problems that incorporate a Hardy potential and a nonlinear convection term. We establish the existence and regularity of solutions under various assumptions concerning the summability of the source term f.

On One Nonlinear Inverse Boundary Value Problem for Linearized Sixth-Order Boussinesq Equation with an Additional Integral Condition

On One Nonlinear Inverse Boundary Value Problem for Linearized Sixth-Order Boussinesq Equation with an Additional Integral Condition

Cross-Diffusion and Higher-Order Chemical Reaction Effects on Hydromagnetic Copper–Water Nanofluid Flow Over a Rotating Cone in a Porous Medium

Spin coating of engineering components with advanced functional nanomaterials which respond to magnetic fields is growing. Motivated by exploring the fluid dynamics of such processes, a mathematical model is developed for chemically reactive Cu–H2O magnetohydrodynamic (MHD) nanofluid swirl coating flow on a revolving vertical electrically insulated cone adjacent to a porous medium under a radial static magnetic field. Heat and mass transfer is included and Dufour and Soret cross-diffusion effects are also incorporated in the model. Thermal and solutal buoyancy forces are additionally included. To simulate chemical reaction of the diffusing species encountered in manufacturing processes, a higher-order chemical reaction formulation is also featured. Via suitable scaling transformations, the governing nonlinear coupled partial differential conservation equations and associated boundary conditions are reformulated as a nonlinear ordinary differential boundary value problem. MATLAB-based shooting quadrature with a Runge–Kutta method is deployed to solve the emerging system. Concentration, temperature and velocity variations for various nondimensional flow parameters have been visualized and analyzed. In addition, key wall characteristics, i.e., radial and circumferential skin friction, Nusselt number and Sherwood number, have also been computed. Validation with earlier studies is also included. The simulations indicate that when compared to a lower-order chemical reaction, a higher-order chemical reaction allows a greater rate of heat and mass transfer at the cone surface. Increasing Dufour (diffuso-thermal) and Soret numbers generally reduces radial and circumferential skin friction and also Nusselt number, whereas it elevates the Sherwood number. Both skin friction components are also suppressed with increasing Richardson number. Strong deceleration in the tangential and circumferential velocity components is induced with greater magnetic field.

On the Existence of Solutions of Nonlinear Boundary Value Problems for Nonshallow Timoshenko-Type Shells with Free Edges

On the Existence of Solutions of Nonlinear Boundary Value Problems for Nonshallow Timoshenko-Type Shells with Free Edges

DYNAMICS OF A SYSTEM OF TWO EQUATIONS WITH A LARGE DELAY

The local dynamics of systems of two equations with delay is considered. The main assumption is that the delay parameter is large enough. Critical cases in the problem of the stability of the equilibrium state are highlighted and it is shown that they have infinite dimension. Methods of infinite-dimensional normalisation were used and further developed. The main result is the construction of special nonlinear boundary value problems which play the role of normal forms. Their nonlocal dynamics determines the behaviour of all solutions of the original system in а neighbourhood of the equilibrium state.

Numerical Approximations of a Class of Nonlinear Second-Order Boundary Value Problems using Galerkin-Compact Finite Difference Method

In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results using the Galerkin weighted residual approach with Bernstein polynomials. The generation of residuals is brought on by the fact that our first approximation is computed using numerical methods. To minimize these residuals, we use the compact finite difference scheme of 4th-order convergence to solve the error differential equations in accordance with the error boundary conditions. We also introduce the formulation of the compact finite difference method of fourth-order convergence for the nonlinear BVPs. The improved approximations are produced by adding the error values derived from the approximations of the error differential equation to the weighted residual values. Numerical results are compared to the exact solutions and to the solutions available in the published literature to validate the proposed scheme, and high accuracy is achieved in all cases.

Construction of a physical-mathematical model of oscillations of the unbalanced vibrator of the pneumatic sorting table

A physical-mathematical model of oscillations of unbalanced vibrators of a pneumatic sorting table as non-stationary oscillations of an impulse-loaded round plate with various options for fixing its contour has been built. The axisymmetric non-stationary oscillations of a round plate supported by a one-sided round base were considered in two ways of fixing its contour, namely, when it is tightly clamped and freely supported. It was assumed that the linearly elastic base resists only compression and does not perceive stretching. It is shown that for certain durations of the transverse force pulse in time, the amplitude of the deflection of the middle of the plate in the direction of action of the external pulse can be smaller than the amplitude of the deflection in the opposite direction. At the same time, in the second case, there is no contact of the plate with the base. It has been proven that this dynamic effect of the asymmetry of the elastic characteristic of the system also applies to bending moments and is more clearly manifested when the contour is freely supported than when it is tightly clamped. For rectangular and sinusoidal pulses, closed-loop solutions of the equations of motion of the plate during its contact with the base and after separation from the base were constructed. Compact formulas were derived for calculating the amplitudes of positive and negative deflections in both directions from the zero position of static equilibrium. Formulas have been obtained for calculating the time it takes for the plate to obtain extreme deflection values, which is achieved due to the selection of a special axisymmetric distribution of dynamic pressure on the plate. Under such a load, the plate simultaneously detaches from the base at all points, except for the contour points, which reduces the nonlinear boundary value problem to a sequence of two linear problems. Numerical integration of the differential equation was carried out to check the reliability of the constructed analytical solutions. Adequacy of the model was proven for the following values of initial parameters: modulus of elasticity, 2.1·1011 Pa; the Poisson ratio of the plate material, 0.25; plate thickness, 7...10 mm; the maximum pressure on the plate, 4·103 Pa; the bending stiffness of the plate, 6402.6667 N·m

A Mathematical Study on a Steady MHD Flow in Double Stratification Medium

The detailed analysis on the flow of MHD fluid in double stratification medium across a stretching sheet with exponential permeability is examined in this research. The approximate analytical solution for the governing equations in steady state is found by using a new approximate analytical method called Ananthaswamy-Sivasankari Method (ASM) and Modified Homotopy Analysis Method (MHAM). The approximate analytical expressions for the dimensionless velocity, dimensionless temperature and dimensionless concentration are derived using these methods. The analytical and numerical results (previous work) are compared and there is a good agreement between our analytical results and numerical works. The impacts of several parameters including porosity, magnetic, suction and heat source parameters are shown in graphical representation. The error table for the physical parameter namely Nusselt number for various values of Prandtl number has been provided. Both ASM and MHAM are very useful to solve some other non-linear boundary value problems especially in MHD fluid flow

A finite difference method on quasi-uniform grids for the fractional boundary-layer Blasius flow

In this paper, we propose a fractional formulation, in terms of the Caputo derivative, of the Blasius flow described by a non-linear two-point fractional boundary value problem on a semi-infinite interval. We develop a finite difference method on quasi-uniform grids, based on a suitable modification of the classical L1 approximation formula and show the consistency, the stability and the convergence. The numerical results confirm the theoretical ones. Comparisons with some recently proposed results are carried out to validate the accuracy of the obtained numerical results, and to show the efficiency and the reliability of the proposed numerical method.

Gravity field modelling in mountainous areas by solving the nonlinear satellite-fixed geodetic boundary value problem with the finite element method

The paper concerns a novel concept based on the iterative approach applied for solving the nonlinear satellite-fixed geodetic boundary value problem (NSFGBVP) using the finite element method (FEM). We formulate the NSFGBVP that consists of the Laplace equation defined in the 3D bounded domain outside the Earth, the nonlinear boundary condition (BC) prescribed on the disretized Earth’s surface, and the Dirichlet BC given on a spherical boundary placed approximately at the altitude of GOCE satellite mission and additional four side boundaries. The iterative approach is based on determining the direction of actual gravity vector together with the value of the disturbing potential. Such a concept leads to the first iteration where the oblique derivative boundary value problem is solved, and the last iteration represents the approximation of the actual disturbing potential and the direction of gravity vector. As a numerical method for our approach, we implement the FEM with triangular prisms. High-resolution numerical experiments deal with the local gravity field modelling in the Andean, Himalayan and Alpine mountain range, where we focus on the contribution to the disturbing potential solution by solving the nonlinear geodetic boundary value problem in comparison with the solution to the oblique derivative geodetic boundary value problem. The obtained results showed the maximal contribution of the presented approach 0.0571 [{text{m}}^{{text{2}}} {text{s}}^{{{text{ - 2}}}}] (Andes), 0.0702 [{text{m}}^{{text{2}}} {text{s}}^{{{text{ - 2}}}}] (Himalayas), 0.0066 [{text{m}}^{{text{2}}} {text{s}}^{{{text{ - 2}}}}] (Alps) in the disturbing potential, that is located in the areas of the highest values of the deflection of vertical.

Quasilinear bessel polynomial collocation method for the solutions of nonlinear singular boundary value problems

The Lane–Emden–Fowler-type equations along with boundary conditions form nonlinear singular boundary value problems. Analytical solutions to this type of problem are not easy, which draws the attention of the researchers toward their numerical solutions. Many methods have been used to provide numerical solutions to this type of problem. In this study, we have proposed the quasilinear Bessel polynomial collocation method (QBPCM) for the solution of these problems. The accuracy and efficiency of the QBPCM have been demonstrated by solving several nonlinear singular models that arise in real-life problems. It has been shown that the numerical results obtained by the proposed method have excellent agreement with the exact solutions and better accuracy than other methods.

New collocation approaches based on reproducing kernel functions for second order nonlinear boundary value problems

In the previous work, the reproducing kernel Hilbert space (RKHS) theory has been employed mainly to solve linear boundary value problems (BVPs). Reproducing kernel functions (RKFs) based techniques for nonlinear BVPs mainly applied iterative techniques. However, there is no error analysis for these methods. In this letter, we shall propose new collocation algorithms based on RKFs for dealing with second order nonlinear BVPs. And the detailed error analysis is provided.

Numerical method of sixth order convergence for solving a fourth order nonlinear boundary value problem

In this paper, we construct an iterative method of sixth order convergence for solving a fourth order nonlinear boundary value problem. The idea of the method is to construct difference schemes of sixth order accuracy for solving Dirichlet problems for second order equation at each iteration of the iterative method on continuous level constructed before by ourselves. Some numerical examples demonstrate the validity of obtained theoretical results.

IDENTIFICATION OF SEPARATION CURVES OF STRONGLY CONTRASTING ENVIRONMENTS USING COMPLEX ANALYSIS METHODS

When modeling mass transfer processes (for example, filtration) in porous media, there may be cases of the existence of highly permeable layers, which are separated from the corresponding studied parts by some curves that must be found (identified) in the process of solving the problem. When constructing a mathematical model of the corresponding physical process, we will consider a highly permeable medium as "ideally (theoretically infinitely) permeable." In this case, the desired curve can be considered an equipotential line. This work considers the stationary process of fluid movement in a homogeneous horizontal layer of infinitely large dimensions - a soil massif, which is limited by infinite sections of curves, in particular by the desired curve of theoretical water resistance and the horizontal axis, on which the local speed of movement is known. On the basis of the methods of conformal reflections and total images, an approach to the identification of the curve of the separation of media is proposed. The constructed algorithm is modified for solving nonlinear inverse boundary value problems on quasi-conformal mappings of curvilinear polygonal regions bounded by uncertain streamlines and equipotential lines. The feature of the method proposed is that the summary representation formulae allow to explicitly write the solution to the localized linear (main) part of the obtained system of equations. The unknown coefficients are determined here by solving a non-linear system generated solely by the boundary conditions and numerical analogues of the Cauchy–Riemann equations.

A CONSTRUCTIVE INVESTIGATION BY THE METHODS OF TWO-SIDED APPROXIMATIONS OF BOUNDARY PROBLEMS FOR SECOND-ORDER SEMILINEAR ELLIPTIC EQUATIONS

The paper considers the first boundary value problem for a second-order semilinear elliptic equation. Problems of this class often arise in the modeling of processes occurring in chemistry, physics, biology, etc. A special place among the methods of analysis of the problems under consideration is occupied by the so-called constructive research methods, which allow not only to prove the existence of a problem solution, but also offer an algorithm for finding it with a given accuracy. For a constructive study of a nonlinear boundary value problem, it is proposed to use two versions of the method of two-sided approximations. Both methods are based on the transition from a differential problem to an equivalent nonlinear integral equation (using the Green’s function or using the Green-Rvachev’s quasi-function), which is analyzed by methods of the theory of nonlinear operators in semi-ordered Banach spaces. Conclusions about the existence of positive solutions of the constructed integral equations and the two-sided convergence of successive approximations to these solutions are made on the basis of the results of V.I. Opojcev on the solvability of nonlinear equations with a heterotone operator. The practical implementation of the method of two-sided approximations based on the use of the Green's function has certain limitations associated with the need to have an explicit expression for this function, which narrows the range of areas in which the method can actually be applied. The method of two-sided approximations, based on the use of the Green-Rvachev’s quasi-function, which can be constructed using the apparatus of the R-functions theory for regions of sufficiently arbitrary geometry, is free from this shortcoming. The proposed methods are illustrated by computational experiments for elliptic equations with Laplace and Helmholtz operators and heterotone power nonlinearity in a number of two- and three-dimensional domains. The results of both methods of two-sided approximations were compared with each other.

Parameter estimates and a uniqueness result for double phase problem with a singular nonlinearity

We consider the boundary value problem −Δpuλ−Δquλ=λg(x)uλ−β in Ω, uλ=0 on ∂Ω with uλ>0 in Ω. We assume Ω is a bounded open set in RN with smooth boundary, 1<p<q<∞, β∈[0,1), g is a positive weight function and λ is a positive parameter. We derive an estimate for uλ which describes its exact behavior when the parameter λ is large. In general, by invoking appropriate comparison principles, this estimate can be used as a powerful tool in deducing the existence, non-existence and multiplicity of positive solutions of nonlinear elliptic boundary value problems. Here, as an application of this estimate, we obtain a uniqueness result for a nonlinear elliptic boundary value problem with a singular nonlinearity.

Numerical method to solve generalized nonlinear system of second order boundary value problems: Galerkin approach

In this study, we consider the system of second order nonlinear boundary value problems (BVPs). We focus on the numerical solutions of &#x0D; different types of nonlinear BVPs by Galerkin finite element method (GFEM). First of all, we originate the generalized formulation of GFEM for those type of problems. Then we determine the approximate solutions of a couple of second order nonlinear BVPs by GFEM. The approximate results are unfolded in tabuler form and portrayed graphically along with the exact solutions. Those results demonstrate the applicability, compatibility and accuracy of this scheme.

A generalized CAS wavelet method for solving ψ-Caputo fractional differential equations

PurposeThe purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem.Design/methodology/approachThe authors have introduced the new generalized operational matrices for the psi-CAS (Cosine and Sine) wavelets, and these matrices are successfully utilized for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem. For the nonlinear problems, the authors merge the present method with the quasilinearization technique.FindingsThe authors have drived the orthogonality condition for the psi-CAS wavelets. The authors have derived and constructed the psi-CAS wavelets matrix, psi-CAS wavelets operational matrix of psi-fractional order integral and psi-CAS wavelets operational matrix of psi-fractional order integration for psi-fractional boundary value problem. These matrices are successfully utilized for the solutions of psi-Caputo fractional differential equations. The purpose of these operational matrices is to make the calculations faster. Furthermore, the authors have derived the convergence analysis of the method. The procedure of implementation for the proposed method is also given. For the accuracy and applicability of the method, the authors implemented the method on some linear and nonlinear psi-Caputo fractional initial and boundary value problems and compare the obtained results with exact solutions.Originality/valueSince psi-Caputo fractional differential equation is a new and emerging field, many engineers can utilize the present technique for the numerical simulations of their linear/non-linear psi-Caputo fractional differential models. To the best of the authors’ knowledge, the present work has never been introduced and implemented for psi-Caputo fractional differential equations.

Existence theory for nabla fractional three-point boundary value problems via continuation methods for contractive maps

In this article, we analyse an $\alpha$-th order, $1 &lt; \alpha \leq 2$, nabla fractional three-point boundary value problem (BVP). We construct the Green's function associated to this problem and derive a few of its important properties. We then establish sufficient conditions on existence and uniqueness of solutions for the corresponding nonlinear BVP using the modern ideas of continuation methods for contractive maps. Our results extend recent results on nabla fractional BVPs. Finally, we provide an example to illustrate the applicability of main results.

An optimal computational method for a general class of nonlinear boundary value problems

An optimal computational method for a general class of nonlinear boundary value problems