Nonlinear Boundary Value Problems Research Articles

Positive solutions for nonlinear fractional boundary value problems

Positive solutions for nonlinear fractional boundary value problems

Solvability of the Quaternary Continuous Classical Boundary Optimal Control Dominated by Quaternary Parabolic System

The purpose of this paper is to study the solvability of the quaternary continuous classical boundary optimal control vector problem dominated by quaternary nonlinear parabolic boundary value problem with state constraints. The existence theorem for a quaternary continuous classical boundary optimal control vector with equality and inequality state constraints is stated and demonstrated under suitable assumptions. The mathematical formulation of the quaternary adjoint eqs. associated with the quaternary nonlinear parabolic boundary value problem with state constraints is discovered. The Fréchet derivative of the cost function and the constraints functions are derived. The necessary and sufficient theorems (conditions) for optimality of the quaternary continuous classical boundary optimal control vector problem are stated and demonstrated under suitable assumptions.

Electromagnetohydrodynamic (EMHD) convective transport of a reactive dissipative carreau fluid with thermal ignition in a non-Darcian vertical duct

A mathematical model is developed to simulate the steady, laminar exothermic reactive electromagneto-hydrodynamic combustible non-Newtonian natural convective transport in a vertical duct. Static uniform axial electrical field and transverse magnetic field are imposed. The Frank-Kamenetskii thermal explosion theory is utilized and also the Carreau fluid model, the latter due to its ability to simulate shear-thinning, Newtonian and shear-thickening behaviour. The duct contains a homogenous, isotropic porous medium and to accomodate Forchheimer inertial drag effects, a non-Darcian model is deployed. The duct walls are permeable enabling suction and injection effects to be studied. Viscous and Joule heating (Ohmic dissipation) are also featured in the model. Following a scaling transformation, the dimensionless emerging non-linear ordinary differential boundary value problem is solved with a robust numerical method (Mathematica shooting algorithm). Validation with an Adomian decomposition method (ADM) is included.

Residual-based error corrector operator to enhance accuracy and reliability of neural operator surrogates of nonlinear variational boundary-value problems

This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. Neural operators have several challenges, including the issue of generating appropriate training data, cost-accuracy trade-offs, and nontrivial hyperparameter tuning. The unpredictability of the accuracy of neural operators impacts their applications in downstream problems of inference, optimization, and control. A framework based on the linear variational problem that gives the correction to the prediction furnished by neural operators is considered based on earlier work in JCP 486 (2023) 112104. The operator, called Residual-based Error Corrector Operator or simply Corrector Operator, associated with the corrector problem is analyzed further. Numerical results involving a nonlinear reaction–diffusion model in two dimensions with PCANet-type neural operators show almost two orders of increase in the accuracy of approximations when neural operators are corrected using the correction scheme. Further, topology optimization involving a nonlinear reaction–diffusion model is considered to highlight the limitations of neural operators and the efficacy of the correction scheme. Optimizers with neural operator surrogates are seen to make significant errors (as high as 80 percent). However, the errors are much lower (below 7 percent) when neural operators are corrected.

Study of a Transmission Problem with Friction Law and Increasing Continuous Terms in a Thin Layer

The aim of this paper is to establish the asymptotic analysis of nonlinear boundary value problems. The non-stationary motion is given by the elastic constructive law. The contact is described with a version of Tresca’s law of friction. A variational formulation of the model, in the form of a coupled system for the displacements and the nonlinear source terms, is derived. The existence of a unique weak solution of the model is established. We also give the problem in transpose form, and we demonstrate different estimates of the displacement and of the source term independently of the small parameter. The main corresponding convergence results are stated in the different theorems of the last section.

Computation of Volterra and Fredholm Integro-Differential Nonlinear Boundary Value Problems by a Modified Regula Falsi-Bisection Shooting Approach

A modified Regula Falsi shooting method approach is deployed to solve a set of Volterra and Fredholm integro- nonlinear boundary value problems. Efforts to solve this class of problems with traditional shooting methods have generally failed and the outcomes from most domain based approaches are often plagued by ill conditioning. Our modification is based on an exponential series technique embedded in a shooting bracketing method. For the purposes of validation, we initially solved problems with known closed form solutions before considering those that do not come with this property but are singular, genuinely nonlinear, and are of practical interest. Although in most of these tests, convergence was found to be super linear, the errors decreased monotonically after few iterations. This suggests that the method is robust and can be trusted to yield faithful results and by far surpasses in simplicity various other techniques that have been applied to solve similar problems. In order to buttress the effectiveness and utility of this approach, we display both the graphical and error analysis outcomes. And for each test case, the method can be seen to demonstrate the closeness of numerically generated results to the analytical solutions.

Computational analysis of hybrid nanoliquid flow over an exponentially elongating sheet influenced by oblique Lorentz force with viscous-Joule dissipation: Role of nanoparticle shape factor

This article mainly examines the influences of an oblique Lorentz force on H2O-based hybrid nanoliquid flow over a over a permeable and exponentially elongating surface embedded in a uniform porous medium. The nanoliquid contains Cu and Al2O3 as suspended nanoparticles of varying shapes (spherical, blade and lamina). Further, the significance of viscous-Joule dissipation, nonlinear thermal radiation, and an internal heat source is also considered. Moreover, Navier’s velocity slip and thermal jump conditions are imposed at the surface boundary. Studying such a fluid flow problem becomes essential because of various industrial devices and technological processes, viz., the production of polymer, manufacturing of food products, metallurgical processes, cooling of electronic chips and solar cells, nuclear power reactors and several other devices like thermistors, glass fiber, electric fuses, gas turbines, etc. A suitable similarity transformation has been utilized to transform the governing set of equations into dimensionless form. The resulting highly nonlinear boundary value problem is solved numerically using Runge–Kutta–Fehlberg formula-based shooting method. In addition, secant iteration is employed to modify the guesses for missing initial conditions. The outcomes of the study are demonstrated and displayed through graphs and tables. Furthermore, a quadratic regression analysis is carried out to illustrate the dominant behavior of the relevant factors on the local drag coefficient and Nusselt number. The angle of inclination of applied magnetic field has the tendency to accelerate the flow. Compared to blade and lamina shapes nanoparticles, nanofluid temperature is higher for spherical nanoparticles.

Simulating accurate and effective solutions of some nonlinear nonlocal two-point BVPs: Clique and QLM-clique matrix methods

This research work deals with two spectral matrix collocation algorithms based on (novel) clique functions to solve two classes of nonlinear nonlocal elliptic two-points boundary value problems (BVPs) arising in diverse physical models. The nonlinearity together with nonlocality makes the models so difficult to solve numerically. In both matrix methods by expanding the unknown solutions in terms of clique polynomials the nonlocality in the models is handled. In the first and direct technique, the clique coefficients are determined in an accurate way through solving an algebraic system of nonlinear equations. To get rid of the nonlinearity of the models and in order to gain efficacy, the quasilinearization method (QLM) is utilized in the second approach. In the latter algorithm named QLM-clique procedure, the first directly clique collocation method is applied to a family of linearized equations in an iterative manner. In particular, we show that the series expansion of clique polynomials is convergent exponentially in a weighted L2 norm. Numerous numerical simulations validate the performance of two matrix collocation schemes and demonstrate the accuracy as well as the gain in computational efficiency in terms of elapsed CPU time. The proposed matrix algorithms for computation are easy to implement and straightforward, and provide more accuracy compared to other available computational values in the literature.

Can we really solve an arch stability problem?

We bring attention to the problem of solving nonlinear boundary-value problems for elastic structures such as arches and shells. Here we discuss a classical problem of a shear-deformable arch postbuckling. Considering a postbuckling behaviour of a circular arch we discuss the possibility to find numerically a solution for highly nonlinear regimes. The main attention is paid to the problem of determination of all solutions. The main conclusion that there is no guarantee to find all solutions, in general.

Multipoint boundary value problem for a coupled system of psi-Hilfer nonlinear implicit fractional differential equation

This study examines the existence and uniqueness of the solution to the coupled system of the ψ-Hilfer nonlinear implicit fractional multipoint boundary value problem. The uniqueness is shown by the Banach contraction principle, and the existence is shown by Krasnosel’skii’s fixed point theorem in a special working space. An example is presented to verify our results. The existence and uniqueness of the solution are analysed graphically.

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 R n ( n ⩾ 3 ) . Our arguments are based on potential theory associated to the polyharmonic operator ( − Δ ) m , where m is a positive integer less than n 2 , properties of functions in the Kato class K m , n , 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.