Nonlinear Boundary Value Problems Research Articles

A Third-order Two Stage Numerical Scheme and Neural Network Simulations for SEIR Epidemic Model: A Numerical Study

This study focuses on the cutting-edge field of epidemic modeling, providing a comprehensive investigation of a third-order two-stage numerical approach combined with neural network simulations for the SEIR (Susceptible-Exposed-Infectious-Removed) epidemic model. An explicit numerical scheme is proposed in this work for dealing with both linear and nonlinear boundary value problems. The scheme is built on two grid points, or two time levels, and is third-order. The main advantage of the scheme is its order of accuracy in two stages. Third-order precision is not only not provided by most existing explicit numerical approaches in two phases, but it also necessitates the computation of an additional derivative of the dependent variable. The proposed scheme's consistency and stability are also examined and presented. Nonlinear SEIR (susceptible-exposed-infected-recovered) models are used to implement the scheme. The scheme is compared with the non-standard finite difference and forward Euler methods that are already in use. The graph shows that the plan is more accurate than non-standard finite difference and forward Euler methods that are already in use. The solution obtained is then looked at through the lens of the neural network. The neural network is trained using an optimization approach known as the Levenberg-Marquardt backpropagation (LMB) algorithm. The mean square error across the total number of iterations, error histograms, and regression plots are the various graphs that can be created from this process. This work conducts thorough evaluations to not only identify the strengths and weaknesses of the suggested approach but also to examine its implications for public health intervention. The results of this study make a valuable contribution to the continuously developing field of epidemic modeling. They emphasize the importance of employing modern numerical techniques and machine learning algorithms to enhance our capacity to predict and effectively control infectious diseases. Doi: 10.28991/ESJ-2024-08-01-023 Full Text: PDF

On Approximate Solutions of a Nonlinear Periodic Boundary-Value Problem with Switching in the Case of Parametric Resonance by the Newton–Kantorovich Method

On Approximate Solutions of a Nonlinear Periodic Boundary-Value Problem with Switching in the Case of Parametric Resonance by the Newton–Kantorovich Method

On the reduction of an autonomous nonlinear boundary value problem unsolved with respect to the derivative to the first-order critical case

On the reduction of an autonomous nonlinear boundary value problem unsolved with respect to the derivative to the first-order critical case

Existence and multiplicity of solutions of Stieltjes differential equations via topological methods

In this work, we use techniques from Stieltjes calculus and fixed point index theory to show the existence and multiplicity of solution of a first order non-linear boundary value problem with linear boundary conditions that extend the periodic case. We also provide the Green’s function associated to the problem as well as an example of application.

Application of the reproducing kernel method for solving linear Volterra integral equations with variable coefficients

This article proposes a new approach for solving linear Volterra integral equations with variable coefficients using the Reproducing Kernel Method (RKM). This method eliminates the need for the Gram-Schmidt process. However, the accuracy of RKM is influenced by various factors, including the selection of points, bases, space, and implementation method. The main objective of this article is to introduce a generalized method based on the Reproducing Kernel, which is successful in solving a special type of singular weakly nonlinear boundary value problems (BVPs). The easy implementation, elimination of the Gram-Schmidt process, fewer calculations, and high accuracy of the present method are interesting. The conformity of numerical results, including tables and figures, with theorems related to error analysis and convergence order, confirms the practicality of the present method.

Equilibrium positions of nonlinear integral-differential boundary value problems unsolved with respect to the derivative

The study of the differential-algebraic boundary value problems was established in the papers of K. Weierstrass, M.M. Lusin and F.R. Gantmacher. Works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boichuk, A. Ilchmann and T. Reis are devoted to the systematic study of differential-algebraic boundary value problems. At the same time, the study of differential-algebraic boundary-value problems is closely related to the study of linear boundary-value problems for ordinary differential equations, initiated in the works of A. Poincare, A.M. Lyapunov, M.M. Krylov, M.M. Bogolyubov, I.G. Malkin, A.D. Myshkis, E.A. Grebenikov, Yu.A. Ryabov, Yu.A. Mitropolsky, I.T. Kiguradze, A.M. Samoilenko, M.O. Perestyuk and O.A. Boichuk. The study of the linear differential-algebraic boundary value problems is connected with numerous applications of corresponding mathematical models in the theory of nonlinear oscillations, mechanics, biology, radio engineering, the theory of the motion stability. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A.M. Samoilenko and O.A. Boichuk on the nonlinear boundary value problems for the integral-differential boundary value problem unsolved with respect to the derivative, in particular, finding the necessary and sufficient conditions of the existence of the desired solutions of the nonlinear integral-differential boundary value problem unsolved with respect to the derivative. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the nonlinear integral-differential boundary value problem unsolved with respect to the derivative in the form of equilibrium positions. The case of a nonlinear equation whose dimension does not coincide with the dimension of the unknown has been researched.

Deformations of Single-Crystal Silicon Circular Plate: Theory and Experiment

In this paper, the experimental methodology for the single-crystal circular plate deformation measurement and subsequent procedure for the quantitation of its mechanical properties are developed. The procedure is based on a new numerical-analytical solution of non-linear boundary-value problem for finite deformations of a circular anisotropic plate. Using the developed method, a study of the deformation of single-crystal circular plates formed on the basis of a silicon-on-insulator structure was carried out. The values of residual stresses are determined and it is shown that the presence of these stresses increases the flexural rigidity of the plate by several times.

Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

In this paper, some one-step iterative schemes with memory-accelerating methods are proposed to update three critical values f′(r), f″(r), and f‴(r) of a nonlinear equation f(x)=0 with r being its simple root. We can achieve high values of the efficiency index (E.I.) over the bound 22/3=1.587 with three function evaluations and over the bound 21/2=1.414 with two function evaluations. The third-degree Newton interpolatory polynomial is derived to update these critical values per iteration. We introduce relaxation factors into the Dzˇunic´ method and its variant, which are updated to render fourth-order convergence by the memory-accelerating technique. We developed six types optimal one-step iterative schemes with the memory-accelerating method, rendering a fourth-order convergence or even more, whose original ones are a second-order convergence without memory and without using specific optimal values of the parameters. We evaluated the performance of these one-step iterative schemes by the computed order of convergence (COC) and the E.I. with numerical tests. A Lie symmetry method to solve a second-order nonlinear boundary-value problem with high efficiency and high accuracy was developed.

Fractional Boundary Layer Flow: Lie Symmetry Analysis and Numerical Solution

In this paper, we present a fractional version of the Sakiadis flow described by a nonlinear two-point fractional boundary value problem on a semi-infinite interval, in terms of the Caputo derivative. We derive the fractional Sakiadis model by substituting, in the classical Prandtl boundary layer equations, the second derivative with a fractional-order derivative by the Caputo operator. By using the Lie symmetry analysis, we reduce the fractional partial differential equations to a fractional ordinary differential equation, and, then, a finite difference method on quasi-uniform grids, with a suitable variation of the classical L1 approximation formula for the Caputo fractional derivative, is proposed. Finally, highly accurate numerical solutions are reported.

Shear Waves in an Elastic Plate with a Hole Resting on a Rough Base

The article is devoted to the analytical and numerical study of the pattern of propagation and attenuation, due to Coulomb friction, of shear waves in an infinite elastic thin plate with a circular orifice of radius r0 lying on a rough base. Considering the friction forces and their influence on the sample of wave propagation in extended rods or thin plates is important for calculating the stress–strain state in them and the size of the area of motion. An exact analytical solution of a nonlinear boundary value problem for tangential stresses and velocities is obtained in quadratures by the Laplace transform, with respect to time. It turned out that the complete exhaustion of the wave front of a strong rupture occurs at a finite distance r* from the center of the orifice, and an elementary formula is given for this distance (the case of tangential shock stresses suddenly applied to the orifice boundary is considered). For various ratios of the magnitude of the limiting friction force to the amplitude of the applied load, the stopping (trailing) wave fronts are calculated. After passing them, a state of static equilibrium between the elastic and friction forces with a nonlinear distribution of residual stresses is established in the region r0≤r≤r*. For the first time, a precise analytical solution was obtained for the boundary value problem of the propagation of elastic shear waves in an infinite isotropic space with a cylindrical cavity, when a tangential shock load is set on its surface.

Mathematical models for analysis of temperature regimes in vehicle braking systems

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in the braking systems of vehicles, which are geometrically depicted as isotropic spatial heat-active media that are subject to internal local thermal heating, have been developed. With the use of classical methods, it is not possible to obtain analytical solutions of linear and nonlinear boundary value problems of mathematical physics in a closed form. This is especially the case when the right-hand sides of differential equations with partial derivatives and boundary conditions are discontinuous functions. The given approach is based on the application of the apparatus of generalized functions to describe the local concentration of thermal influence. This made it possible to apply the integral transformation and, on this basis, to obtain analytical solutions of both linear and nonlinear boundary value problems. In the case of a nonlinear boundary value problem, the Kirchhoff transformation was applied, using which the original nonlinear heat conduction equation and nonlinear boundary conditions were linearized, and as a result, a linearized second-order differential equation with partial derivatives and boundary conditions with a discontinuous right-hand side were obtained. To solve the linear boundary value problem, as well as the obtained linearized boundary value problem with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, an analytical relationship was obtained for determining the temperature distribution in this medium. On the basis of the developed mathematical models, a computational algorithm was created and on this basis, software tools were created, using which the heat exchange processes in the middle of the brake structures for the selected materials of the brake pads were analyzed in terms of their effectiveness, as well as the determination of the optimal temperature values for the effective operation of the braking system of vehicles. The developed linear and nonlinear mathematical models for determining the temperature field in spatial heat-active media with internal heating make it possible to analyze their thermal stability. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and individual elements, but also the entire structure.

An explicit Jacobian for Newton's method applied to nonlinear initial boundary value problems in summation-by-parts form

<p>We derived an explicit form of the Jacobian for discrete approximations of a nonlinear initial boundary value problems (IBVPs) in matrix-vector form. The Jacobian is used in Newton's method to solve the corresponding nonlinear system of equations. The technique was exemplified on the incompressible Navier-Stokes equations discretized using summation-by-parts (SBP) difference operators and weakly imposed boundary conditions using the simultaneous approximation term (SAT) technique. The convergence rate of the iterations is verified by using the method of manufactured solutions. The methodology in this paper can be used on any numerical discretization of IBVPs in matrix-vector form, and it is particularly straightforward for approximations in SBP-SAT form.</p>

Unified existence results for nonlinear fractional boundary value problems

<abstract><p>In this work, we focus on investigating the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. By extending the framework of the technique based on well-ordered coupled lower and upper solutions, we guarantee the existence of solutions in a sector defined by these solutions. One notable aspect of our study is that the proposed approach unifies the existence results for the problems that have previously been discussed separately in the literature. To substantiate these findings, we have added three illustrative examples.</p></abstract>

Spectral approach with iterative clarification of a radiation boundary conditions for modeling of quasimodes of a gyrotrons open cavities

Purpose. The article presents a new method for numerical simulation of quasi-eigenmode oscillations in open resonators of gyrotrons — powerful vacuum generators of electromagnetic waves in the millimeter and submillimeter ranges. The gyrotron cavity has the shape of a weakly inhomogeneous hollow circular metal waveguide. Methods. The proposed approach uses the inhomogeneous string equation with radiation boundary conditions to formulate a nonlinear spectral boundary value problem describing oscillations in a resonator, neglecting the couplings of waves with different radial indices. By linearizing with respect to frequency the radiation boundary conditions, the boundary value problem is reduced to a linear boundary value problem. To discretize this boundary value problem, the finite difference method is used and a linear generalized matrix eigenvalue problem is formulated. This problem is solved by the Arnoldi method with eigenvalues calculation in a shift-invert mode. An iterative algorithm is proposed that makes it possible to sequentially calculate a given number of frequencies and quality factors of quasi-eigenmodes of oscillations. Results. The computer program was developed written in the Wolfram Language and Fortran using the algorithms proposed in the work. The results of test calculations for real gyrotron resonators are presented, which demonstrate the high accuracy of the obtained values of frequencies, quality factors and field distributions of quasi-eigenmode oscillations in the studied resonators. Conclusion. The methods, algorithms and created program proposed in the article can significantly facilitate the process of developing gyrotrons intended for various practical applications and operating in new frequency ranges. The method of iterative refinement of boundary conditions can be generalized to the case of equations of the linear theory of a gyrotron and used to develop new methods for analyzing the starting conditions for the soft self-excitation in gyrotrons — generators.

Existence of nodal solutions of nonlinear Lidstone boundary value problems

Existence of nodal solutions of nonlinear Lidstone boundary value problems

Fractional differential equation with movable boundary conditions

In this research paper, we discuss the complex-valued solutions for the nonlinear fractional boundary value problem (FBVP) of complex order (δ = τ + ιa; 1 < τ ≤ 2, a ∈ R+) with movable boundary conditions. The fractional operators are taken in the sense of Riemann-Liouville (R-L) with complex order. By using the concept of Green’s function, the existence and uniqueness of solutions are established in this article. Also, we prove that the FBVP of complex order with movable boundary conditions is Ulam-Hyers Stable. Using illustrative examples, the results for this nonlinear FBVP have been shown.

Solutions of initial and boundary value problems using invariant curves

<p>The purpose of this study is to investigate the solutions of initial and boundary value problems of ordinary differential equations by employing Lie symmetry generators. In this investigation, it shown that invariant curves, which obtained by symmetry generators, also be utilized to find solutions to initial and boundary value problems. A method, involving invariant curves, presented to find solutions to initial and boundary value problems. Solutions to many linear and nonlinear initial and boundary value problems discussed by applying the proposed method.</p>

Solutions and anti-periodic solutions for impulsive differential equations and inclusions containing Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $ in infinite dimensional Banach spaces

<abstract><p>In this paper, we improved recent results on the existence of solutions for nonlinear fractional boundary value problems containing the Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $. We also derived the exact relations between these fractional boundary value problems and the corresponding fractional integral equations in infinite dimensional Banach spaces. We showed that the continuity assumption on the nonlinear term of these equations is insufficient, give the derived expression for the solution, and present two results about the existence and uniqueness of the solution. We examined the case of impulsive impact and provide some sufficiency conditions for the existence and uniqueness of the solution in these cases. We also demonstrated the existence and uniqueness of anti-periodic solution for the studied problems and considered the problem when the right-hand side was a multivalued function. Examples were given to illustrate the obtained results.</p></abstract>

Positive solutions of a nonlinear three-point p-Laplacian fractional boundary value problem with infinitely many singularities

In this paper, an attempt has been made to establish the existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential equation subject to the non-local boundary conditions. We have shown that there exist countably many positive solutions with the help of fixed point index theory in a cone on a Banach space.

Solvability of nonlinear boundary value problems for non-sloping Timoshenko-type isotropic shells of zero principal curvature

We study the solvability of boundary value problem for a system of second order partial differential equations under boundary given conditions describing the equilibrium of elastic non-sloping isotropic inhomogeneous shells with free boundary in the framework of the Timoshenko shear model. The base of the study method are the integral representations for generalized motions involving arbitrary functions, which also involve arbitrary holomorphic functions. The arbitrary functions ate determined so that the generalized motions satisfy a linear system of equations and linear boundary conditions extracted from the original boundary value problem. The holomorphic functions are sought as Cauchy type integrals with real densities. The integral representations allow us to reduce the initial boundary value problem to a nonlinear operator equations for generalized motions in the Sobolev space. While studying the solvability of this operator equation, the most essential point is to invert it with respect to the linear part. As a result, the work is reduced to an equation, the solvability of which is established on the base of the contracting mapping principle.