Nonlinear Boundary Value Problems Research Articles

Modified α -Parameterized Differential Transform Method for Solving Nonlinear Generalized Gardner Equation

In this article, we present a novel enhancement to the α -parameterized differential transform method (PDTM) for solving nonlinear boundary value problems. The proposed method is applied to solve the generalized Gardner equation by utilizing genetic algorithms to obtain optimal parameter values. Our proposed approach extends the general differential transformation method, allowing for the use of various values for the coefficient α . Our solution procedure offers a distinct advantage by allowing the original differential transformation method to be divided into multiple steps, thereby illustrating specific solution properties for nonlinear boundary value problems. Additionally, possible alternative solutions based on varying parameter values are also explored and discussed. The results with those obtained through the DTM method and exact solutions are compared to confirm the accuracy of our method and its efficiency in reaching the exact solution quickly.

Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

This paper investigates an effective computational method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems appeared in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica®12. The improved effective computational methods (I-ECMs) have been implemented to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the proposed methods has been presented. Furthermore, the maximum error remainder () has been computed to prove the proposed methods' accuracy. The results convincingly prove that ECM and I-ECMs are effective and accurate in obtaining novel approximate solutions to the problems.

Nonlinear polyharmonic boundary value problems in the punctured unit ball

In this paper we investigate the existence and the asymptotic behavior of positive continuous solutions for a class of nonlinear polyharmonic boundary value problems in the punctured unit ball of . Our arguments are based on potential theory associated to the polyharmonic operator , where m is a positive integer less than , properties of functions in the Kato class , Karamata regular variation theory tools and the Schauder fixed point theorem.

Stretching/shrinking impact on the transient heat transfer in a radial moving porous fin of a longitudinal trapezoidal structure

The prime goal of the current pagination is to scrutinize the transient thermal response in a radial porous moving fin of a longitudinal trapezoidal structure. The solution of the constructed problem is heeded by a pdepe solver in Matlab, which works on the centered-finite difference technique. Darcy’s model is utilized to investigate porous nature. A stretching/shrinking mechanism has been incorporated to improve the fin efficiency. Stretching reduces efficiency, whereas shrinking improves fin efficiency indicating major benefits of fin design consideration. The relevant temperature results are shown visually and discussed quantitatively in detail to evaluate the competence of the suggested fin. With a rise in the Peclet number, temperature distribution decreases. A similar nature is noted with respect to the power index of convective heat transfer coefficient, convection and radiation parameters, porous parameter, stretching/shrinking parameter, and thermal conductivity gradient with temperature. The graphical outcomes also show that the fin temperature amplifies with the ambient temperature parameter. Also, it is noticed that thermal distribution is a decreasing function of the power index of the convective heat transfer coefficient such that there is a decrease of 1.64% in temperature distribution as the power index increases. To support the validity of the solutions, a comparison was made with notable results from the existing literature for the specific case of this study. The novelty of the work is to study highly nonlinear boundary value problems and such nonlinearities are more relevant in the system that describes the thermal processing of materials. It is also noted that initial transient heat transfer vanishes quickly under steady-state conditions when the material is exposed to the ambient temperature.

Математичні моделі температурного поля у елементах електронних пристроїв із чужорідним включенням

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic media with semi-transparent foreign inclusions that are also subject to external heat load, have been developed. To solve the nonlinear boundary-value problem, a linearizing function was introduced, using which the original nonlinear heat conduction equation and nonlinear boundary conditions were linearized, and as a result, a partially linearized second-order differential equation with partial derivatives and discontinuous and singular coefficients relative to the linearizing function and partially linearized boundary conditions was obtained. For the final linearization of the partially linearized differential equation and boundary conditions, the approximation of the temperature according to one of the spatial coordinates on the boundary surfaces of the inclusion was performed by piecewise constant functions. To solve the obtained linear boundary value problem, the Hankel integral transformation method was used, as a result of which an analytical solution was obtained, which determines the introduced linearizing function. For the numerical analysis of temperature behavior and heat exchange processes caused by external heat load, software tools have been developed, which are used to create a geometric image of temperature distribution depending on spatial coordinates. The obtained results indicate the correspondence of the developed mathematical models of the analysis of heat exchange processes in heterogeneous media with external heating to a real physical process. The obtained results indicate the correspondence of the developed mathematical models of the analysis of heat exchange processes in heterogeneous media with external heating to a real physical process. The developed models make it possible to analyze environments with foreign elements under external thermal loads regarding their thermal resistance. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual elements, but also the entire structure.

Computation of couple stress electroconductive polymer from an exponentially stretching sheet

Magnetic polymer processing involves multiple physical phenomena and requires simultaneous consideration of rheological, hydromagnetic and thermal characteristics. Inspired by new developments in functional magnetic coating dynamics, the present article investigates the viscous magnetohydrodynamic non-Newtonian boundary layer flow of an incompressible, electrically conducting, couple stress, electroconductive polymer from an exponentially stretching sheet. The nonlinear boundary value problem is solved numerically by adopting shooting technique and bvp4c algorithm available in MATLAB software. Validation with the Adams-Moulton 2-step predictor corrector algorithm is included. The effects of non-Newtonian couple stress parameter, magnetic interaction number, mixed convection parameter, slip parameter, Prandtl and Eckert number on the primary and secondary velocities and temperature fields are visualized graphically. Greater hydrodynamic slip effect initially boosts the primary velocity near the wall but thereafter induces a decrement in it. However, a consistent improvement in secondary velocity is computed for greater slip parameter. Increasing Hall parameter strongly accelerates the secondary flow but only weakly accelerates the primary flow. Increasing magnetic interaction number decelerates the primary and secondary flow. Elevating magnetic number promotes the thermal boundary layer thickness and also the temperature. Couple stress effects generally accelerate both the primary and secondary flow.

Dynamics of Chains of Many Oscillators with Unidirectional and Bidirectional Delay Coupling

Chains of Van der Pol equations with a large delay in coupling are considered. It is assumed that the number of chain elements is also sufficiently large. In a natural manner, a chain is replaced by a Van der Pol equation with an integral term in the space variable and with periodic boundary conditions. Primary attention is given to the local dynamics of chains with unidirectional and bidirectional coupling. For sufficiently large values of the delay parameter, parameters are explicitly determined for which critical cases occur in the stability problem for the zero equilibrium state. It is shown that the problems under consideration have an infinite-dimensional critical case. The well-known methods of invariant integral manifolds and the methods of normal forms are inapplicable in these problems. Proposed by this paper’s author, the method of infinite normalization—the method of quasi-normal forms—is used to show that the leading terms of the asymptotics of the original system are determined by solutions of (nonlocal) quasi-normal forms, i.e., special nonlinear boundary value problems of the parabolic type. As the main results, corresponding quasi-normal forms are constructed for the considered chains.

Quaternary Boundary Optimal Control Problem Dominating by Quaternary Nonlinear Parabolic System

In this paper, our purpose is to study the quaternary continuous classical boundary optimal control vector problem (QCCBOCVP) dominated by the quaternary nonlinear parabolic boundary value problem (QNLPBVP). Under suitable assumptions and with given quaternary continuous classical boundary control vector (QCCBCV), the existence theorem for a unique quaternary state vector solution (QSVS) of the weak form (WF) for the QNLPBVP is stated and demonstrated via the Method of Galerkin and the first compactness theorem. Furthermore, the continuity of the Lipchitz operator between the QSVS of the WF for the QLPBVP and the corresponding QCCBCV is proved. The existence of a quaternary continuous classical boundary optimal control vector (QCCBOVC) is stated and demonstrated under suitable assumptions.

Non-local fractional boundary value problems with applications to predator-prey models

We study a nonlinear fractional boundary value problem (BVP) subject to non-local multipoint boundary conditions. By introducing an appropriate parametrization technique we reduce the original problem to an equivalent one with already two-point restrictions. Using a notion of Chebyshev nodes and Lagrange polynomials we construct a successive iteration scheme, that converges to the exact solution of the non-local problem for particular values of the unknown parameters, which are calculated numerically.
 For mote information see https://ejde.math.txstate.edu/Volumes/2023/58/abstr.html

Thermal radiation influence on non-Newtonian nanofluid flow along a stretchable surface with Newton boundary condition

This article focuses on the heat transfer on a mixed convection flow of nanoparticles suspended non-Newtonian nanofluid past a stretching surface under Newtonian boundary conditions. The leading boundary layer equations are changed to a non-linear boundary value problem by using suitable similarity variables. BVP4c with MATLAB is used to elucidate the resultant system. The controlled parameters involved in the flow field problem and its impact on the flow components as well as the interested engineering quantities are examined via graphs and tables. Our study results are in good agreement with the existing literature. It is noticed that the presence of radiation increases the heat inside the fluid, which leads to an increase in the thickness of the energy boundary layer and an increase in the temperature of the fluid. The temperature of the fluid decreases for higher values of the Brownian motion parameter and increases for variations in thermophoresis.

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.