Solution Of Boundary Value Problem Research Articles

Multiplicity of Solutions for Discrete 2n-TH Order Periodic Boundary Value Problem with φp-Laplacian

The purpose of this paper is to investigate the existence and multiplicity of nontrivial solutions with the φp-Laplacian for the discrete 2n-th order periodic boundary value issue. To support these conclusions, we have employed variational techniques and contemporary critical point theory. A few new findings are expanded upon and enhanced. We give an example to show how our key findings can be applied.

A novel Picard–Ishikawa–Green's iterative scheme for solving third‐order boundary value problems

The purpose of this paper is to introduce a novel fixed point iterative scheme based on Green's function, called the Picard–Ishikawa–Green's iterative scheme and use it in approximating the solution of boundary value problems (BVPs). It is proved that Picard–Ishikawa–Green's scheme converges strongly for an integral operator which represents the solution of BVP and the scheme is stable. Moreover, we prove that the integral operator is a contraction. Furthermore, it is shown that the novel scheme converges faster than all of Ishikawa–Green's, Khan–Green's, and Mann–Green's schemes. Finally, numerical examples are given to substantiate the validity of our results for third‐order BVPs. Our results extend and generalize several other results in literature.

Eleventh Degree Polynomial Series Solution Approach of Special Non-linear Fourth Order Boundary Value Problems

In this paper a new eleventh degree polynomial series solution approach is developed for the solution of special non-linear fourth order boundary value problems. The method consist first of obtaining a linear differentials system of twelve equations from the boundary conditions, governing equation and its three successive derivatives which were evaluated at boundary points. Secondly we assume the approximate solution in the form of a polynomial of degree eleventh with twelve unknown coefficients. To determine the unknown coefficients we incorporate the assumed solution into linear differentials systems of twelve equations which results into a linear systems of twelve equations with twelve unknown and which can be solve uniquely. It is clear from the tables and figures that the method is in good agreement with the exact and with some existing results in the literatures. Also it can be seen from example 3.3 that the exact solution is reproduced which is an added advantage of the method.

On Graphical Symmetric Spaces, Fixed-Point Theorems and the Existence of Positive Solution of Fractional Periodic Boundary Value Problems

The rationale of this work is to introduce the notion of graphical symmetric spaces and some fixed-point results are proved for H-(ϑ,φ)-contractions in this setting. The idea of graphical symmetric spaces generalizes various spaces equipped with a function which characterizes the distance between two points of the space. Some topological properties of graphical symmetric spaces are discussed. Some fixed-point results for the mappings defined on graphical symmetric spaces are proved. The fixed-point results of this paper generalize and extend several fixed-point results in this new setting. The main results of this paper are applied to obtain the positive solutions of fractional periodic boundary value problems.

An explicit substructuring method for overlapping domain decomposition based on stochastic calculus

In a recent paper [7], a hybrid supercomputing algorithm for elliptic equations has been proposed. The idea is that the interfacial nodal solutions solve a linear system, whose coefficients are expectations of functionals of stochastic differential equations confined within patches of about subdomain size. Compared to standard substructuring techniques, such as the Schur complement method for the skeleton, the hybrid approach produces an explicit and sparse shrunken matrix—hence suitable for substructuring again. The ultimate goal is to push strong scalability beyond the state of the art by leveraging the potential for parallelisation of stochastic calculus. Here, we present a major revamping of that framework, based on the insight of embedding the domain in a cover of overlapping circles (in two dimensions). This allows for efficient Fourier interpolation along the interfaces (now circumferences) and—crucially—for the evaluation of most of the interfacial system entries as the solution of small boundary value problems on a circle. This is both extremely efficient (as they can be solved in parallel and by the pseudospectral method) and free of Monte Carlo error. Stochastic numerics are only needed on the relatively few circles intersecting the domain boundary. In sum, the new formulation is significantly faster, simpler, and more accurate while retaining all of the advantageous properties of PDDSparse. Numerical experiments are included for the purpose of illustration.

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.

On the Solution of Caputo Fractional High-Order Three-Point Boundary Value Problem with Applications to Optimal Control

This research paper establishes the existence and uniqueness of solutions for a non-integer high-order boundary value problem, incorporating the Caputo fractional derivative with a non-local type boundary condition. The analytical approach involves the introduction of the fractional Green’s function. To analyze our findings effectively, we apply the Banach contraction fixed point theorem as the primary principle. Furthermore, we illustrate our results through the presentation of various examples.

Mixed fractional order p-Laplacian boundary value problem with a two-dimensional Kernel at resonance on an unbounded domain

In this paper, by using the Ge and Ren extension of coincidence degree theory, we established the existence of a solution for a resonant mixed fractional order p-Laplacian boundary value problem (BVP) on the half-line. In the process, we solved the corresponding homogeneous fractional order BVP for conditions critical for resonance and showed that the operator A(x,λ)(t) constructed from the abstract equation Mx(t)=Nx(t) is relatively compact. The results are demonstrated with an example.

RESEARCH OF THE STATIONARY PHENOMENON OF HEAT TRANSFER IN A SPHERICAL CONDENSED MEDIA

A method for investigating approximate analytical solutions of boundary value problems of singularly perturbed equations of stationary heat transfer processes in a spherical condensed medium, close to solving the boundary value problem for an undisturbed equation, is presented. The main task in which the structure of the article is defined is the development of mathematical and computer models of the stationary phenomenon of heat transfer in flat condensed media and a model of diffusion-reactive energy transfer depending on the thermal conductivity of materials.

Analysis of Varying Temperature Regimes in a Conductive Strip during Induction Heating under a Quasi-Steady Electromagnetic Field

Transition processes in a steel conductive strip are analyzed during its induction heating under a quasi-steady electromagnetic field. In particular, the temperature field in the strip is studied. A method of solving corresponding initial boundary problems in a two-dimensional mathematical model for differential equations of electrodynamics and heat conduction is developed. The Joule heat and the temperature are determined with a high level of accuracy. The defining functions are the temperature and component of the magnetic field intensity vector tangent to the bases and end planes of the strip. To find them, we use cubic approximation of the defining functions’ distribution along the thickness coordinate. The original two-dimensional initial boundary value problems for the defining functions are reduced to one-dimensional initial boundary value problems on their integral characteristics. General solutions for these problems are obtained using the finite integral transformation by the transverse variable and the Laplace transform of the integral by time. Integral characteristics’ expressions are represented as convolutions for functions that describe homogeneous solutions of one-dimensional initial boundary value problems and limiting values of defining functions on the bases and end planes of the strip. The change of temperature under a varying regime in the dimensionless Fourier time and temperature distribution over the strip cross-section in a steady state depending on the parameters of induction heating and the Biot number are numerically analyzed. Varying and constant temperature regimes of the strip under conditions of the near-surface and continuous induction heating are studied.

Numerical approaches for solution of hyperbolic difference equations on circle

Abstract The present paper considers nonlocal boundary value problems for hyperbolic equations on the circle T 1 \mathbb{T}^{1} . The first-order modified difference scheme for the numerical solution of nonlocal boundary value problems for hyperbolic equations on a circle is presented. The stability and coercivity estimates in various Hölder norms for solutions of the difference schemes are established. Moreover, numerical examples are provided.

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.

Existence Result for Fractional Differential Equation on Unbounded Domain

In this article, we establish certain sufficient conditions to show the existence of solutions of boundary value problem for fractional differential equations on the half-line in a Fréchet space. The main result is based on Tykhonoff fixed point theorem combining with a suitable measure of non-compactness. An example is given to illustrate our approach.

Uniqueness of Solutions for Boundary Value Problems of Nonlinear Coupled Fractional Differential Equations with Integral Boundary Conditions

Uniqueness of Solutions for Boundary Value Problems of Nonlinear Coupled Fractional Differential Equations with Integral Boundary Conditions

Technical and economic aspects of designing a hybrid microwave electrotechnological installation for processing biological substrates and plant raw materials

The paper presents an algorithm for designing a hybrid microwave electrotechnological installation for processing biological substrates and plant material. The feasibility of designing a hybrid microwave electrotechnological installation is considered on the basis of comparative economic effects. A technical and economic optimization of the structure and parameters of a hybrid microwave electrotechnological installation for non-thermal processing of biological substrates and thermal processing of plant material was carried out. Purpose of the study: To conduct a feasibility study for the design of a hybrid microwave electrotechnological installation for processing biological substrates and plant raw materials. Methodology: The design and development of a hybrid microwave electrotechnological installation for thermal processing of plant material and non-thermal processing of biological substrates is based on the solution of a consistent boundary value problem of electrodynamics and heat and mass transfer. Results: The design algorithm and methodology for technical and economic calculation of a hybrid microwave electrotechnological installation for processing biological substrates and plant material are presented.

Nontrivial solutions for a Hadamard fractional integral boundary value problem

<abstract><p>In this paper, we studied a Hadamard-type fractional Riemann-Stieltjes integral boundary value problem. The existence of nontrivial solutions was obtained by using the fixed-point method when the nonlinearities can be superlinear, suberlinear, and have asymptotic linear growth. Our results improved and generalized some results of the existing literature.</p></abstract>

Hybrid microwave electrotechnological installation for processing biological substrates and plant material

The work presents a hybrid microwave electrotechnological installation, in which it is possible to simultaneously process biological substrates and plant material. A general view of the hybrid microwave electrotechnological installation and a mathematical model of the technological unit for thermal microwave processing of plant material are presented. The results of experimental studies of the heat treatment of plant material in a hybrid microwave electrotechnological installation are presented. Purpose of the study: To conduct experimental studies of the processing modes of plant material in a hybrid microwave electrotechnological installation. Methodology: The design and development of a hybrid microwave electrotechnological installation for thermal processing of plant material and non-thermal processing of biological substrates is based on the solution of a consistent boundary value problem of electrodynamics and heat and mass transfer. Results: As a result of experimental studies on the heat treatment of plant material in a hybrid microwave electrotechnological installation, the preferred processing modes were determined, resulting in a uniform change in the temperature and humidity of the plant material and a reduction in processing time.

LYAPUNOV-TYPE INEQUALITY FOR CERTAIN HALF-LINEAR LOCAL FRACTIONAL ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we establish a Lyapunov-type inequality for the half-linear local fractional ordinary differential equation based on the formulation of the local fractional derivative. In addition, we apply the inequality to investigate the non-existence and uniqueness of solutions for related homogeneous and non-homogeneous local fractional 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.

Multiple Positive Solutions of Discrete Fractional Boundary Value Problems

Multiple Positive Solutions of Discrete Fractional Boundary Value Problems