Class Of Boundary Value Problems Research Articles

Stability estimates for some parabolic inverse problems with the final overdetermination via a new Carleman estimate

This paper is about Hölder and Lipschitz stability estimates and uniqueness theorems for some coefficient inverse problems and associated inverse source problems for a general linear parabolic equation of the second order with variable coefficients. The data for the inverse problem are given at the final moment of time . In addition, both Dirichlet and Neumann boundary conditions are given either on a part or on the entire lateral boundary. Thus, if these boundary conditions are given only at a part of the boundary, then even if the target coefficient is known, still the forward problem is not a classical initial boundary value problem.

Existence results for boundary value problems of Hadamard fractional differential equations on unbounded domain

Abstract In this paper we investigate the existence and uniqueness of solutions for a class of boundary value problem for fractional differential equations involving the Hadamard’s fractional derivative, by applying a nonlinear alternative of Leray–Schauder due to Frigon and Granas for contraction maps in Fréchet spaces.

Existence of positive solutions to a class of boundary value problems with derivative dependence on the half-line

Existence of positive solutions to a class of boundary value problems with derivative dependence on the half-line

On Some Impulsive Fractional Integro-Differential Equation with Anti-Periodic Conditions

We investigate a class of boundary value problems (BVPs) involving an impulsive fractional integro-differential equation (IF-IDE) with the Caputo–Hadamard fractional derivative (C-HFD). We employ some fixed-point theorems (FPTs) to study the existence of this fractional BVP and its unique solution. The boundary conditions (BCs) established in this study are of a more general type and can be reduced to numerous specific examples by defining the parameters involved in the conditions. In this way, we extend some recent nice results. At the end, we use an example to verify our results.

A novel continuum dislocation density field-based crystal plasticity theory

In this work, a novel dislocation density field-based crystal plasticity formulation, that incorporates up-scaled continuum dislocation density fields to represent all possible characters of the dislocation density, is presented. The continuum dislocation field theory, formulated assuming large strain kinematics, is based on an all-dislocation concept, whereby individual dislocation density types are described as vector fields. The evolutionary behaviour of the dislocation density fields is defined in terms of a glide component derived from the classical general conservation law for dislocation density fields proposed originally by Kroner and Acharya, a rotation one given by the curl of the dislocation velocity vector as proposed originally by Ngan and co-workers, and a statistically stored dislocation source/sink component.The proposed formulation has been numerically implemented within the finite element method using a modified up-wind stabilisation approach, which results on eight extra independent nodal degrees of freedom per slip system, associated with the upscaled dislocation densities of pure edge and screw character. Furthermore, the required boundary conditions are simple: initial total dislocation density and dislocation flux or velocity, without the need for higher order boundary conditions. Full details of the numerical implementation of the crystal plasticity theory are given.The theory is then used to investigate classical boundary value problems involving 1D and 2D dislocation pile-ups, and the development of boundary layers in a deforming single crystal strip under constrained conditions. The main general understanding that emerges from the case studies is that, as either the dislocation pile-ups or the boundary layers develop with deformation, geometrically necessary dislocations are generated and lead to an overall behaviour that is size-dependent. Furthermore, it is shown that, in the absence of sources or sinks, which are statistically stored in nature, that total number of dislocations in the pile-up problems is preserved. The numerical predictions of the 1D pile-up and strip simple shear problems are in good agreement with published solutions.

On Tempered (κ, ψ)-Hilfer Fractional Boundary Value Problems

Our research is primarily focused on applying the tempered (κ, ψ)-fractional operators to investigate the existence, uniqueness, and κ-Mittag-Leffler-Ulam-Hyers stability of a specific class of boundary value problems involving implicit nonlinear fractional differential equations and tempered (κ, ψ)-Hilfer fractional derivatives. To accomplish this, we make use of the fixed point theorem of Banach and a generalization of the well-known Gronwall inequality. Additionally, we provide illustrative examples to demonstrate the practical effectiveness of our main findings.

Solution approximation of fractional boundary value problems and convergence analysis using AA-iterative scheme

<abstract><p>Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature.</p></abstract>

An investigation of a new Lyapunov-type inequality for Katugampola–Hilfer fractional BVP with nonlocal and integral boundary conditions

In this manuscript, we focus our attention on investigating new Lyapunov-type inequalities (LTIs) for two classes of boundary value problems (BVPs) in the framework of Katugampola–Hilfer fractional derivatives, supplemented by nonlocal, integral, and mixed boundary conditions. The equivalent integral equations of the proposed Katugampola–Hilfer fractional BVPs are established in the context of Green functions. Also, the properties of these Green functions are proved. The LTIs are investigated as sufficient criteria for the existence and nonexistence of nontrivial solutions for the subjected problems. Our systems are more general than in the literature, as a consequence there are many new and known specific cases included. Finally, our results are applied for estimating eigenvalues of two given BVPs.

On a New Class of Impulsive η-Hilfer Fractional Volterra-Fredholm Integro-Differential Equations

This work addresses the idea of the uniqueness and existence results for a class of boundary value problems (BVPs) for implicit Volterra-Fredholm integro-differential equations (V-FIDEs) with fractional η-Hilfer nonlinear equations and multi-point fractional boundary non-instantaneous conditions. The conclusions are confirmed by the fixed point of Krasnoselskii's theorem and the Banach contraction principle. Finally, a concrete example is given to illustrate our main conclusions.

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

PurposeA unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).Design/methodology/approachThe original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.FindingsThe bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.Originality/valueThe symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.

A discrete-based multi-scale modeling approach for the propagation of seismic waves in soils

A three-dimensional multi-scale discrete–continuum model (Finite Volume Method × Discrete Element Method, FVM × DEM) is developed for a discrete-based description of the mechanical behavior of granular soils in boundary value problems (BVPs). In such a scheme, the constitutive response of the material is derived through direct DEM computations on a representative volume element attached to each mesh element. The developed multi-scale approach includes the inertial effect in the stress homogenization formulation and serves to study the mechanism of propagation of seismic waves, in comparison with a more classical BVP simulation that adopts an advanced bounding surface plasticity model “P2PSand”. We start with a detailed and fair calibration and validation of these two models against laboratory tests for Toyoura sand under monotonic and cyclic loading. Then, the performance of the two approaches is compared for the case of a seismic wave loading passing through a saturated soil column with different relative densities, revealing several differences between the results of the two models.

El-Numodis: a new tool to model dislocation and surface interactions

While surfaces are known to have a limited impact on the mechanical properties of crystalline materials at the macroscopic scale, they play a key role at small-scale behaving alternatively as sources or sinks of various plastic deformation processes. In this study, we present a new tool called El-Numodis that relies on the superposition method to couple the discrete dislocation dynamics code Numodis to Elmer, an open-source finite-element-modeling tool. After few years of development, El-Numodis allows now for the simulation of small-scale object deformation and mechanical properties based on a large set of surface-related processes including stress-free boundaries, mirrored dislocations and a Monte-Carlo based dislocation nucleation mechanism. Here we present the main features of the code as well as numerical test-cases and benchmarks going from classical boundary value problems to tensile tests on model thin film.

On qualitative analysis of boundary value problem of variable order fractional delay differential equations

Variable order differential equations are the natural extension of the said area. In many situations, such problems have the ability to describe real-world problems more concisely. Therefore, keeping this validity in mind, we have considered a class of boundary value problems (BVPs) under the variable order differentiation. For the suggested problems, we have developed the existence and uniqueness (EU) by using some fixed point results due to Banach and Schauder. Sufficient adequate results have been established for the required need. Some stability results have also been elaborated based on the concepts of Ulam, Hyers, and Rassias. Proper examples have also been provided with detailed analysis to verify our results.

Destabilization of synchronous periodic solutions for patch models

We study the destabilization of synchronous periodic solutions for general patch-models with cross-diffusion-like couplings, where the underlying kinetic systems have stable periodic solutions bifurcating from double homoclinic loops. We first reduce the stability problem of synchronous periodic solutions for patch models into that for lower-dimensional systems, then analyze the destabilization by using the Floquet theory and solving a class of boundary value problems. After establishing the Fredholm alternative properties for an auxiliary linear operator, we give the characteristic function to determine the Floquet spectra for the reduced systems, and then the conditions for the destabilization. Finally, we apply the main results to a patch model with a two-dimensional kinetic system.

Data-driven computational framework for snap-through problems

The distance-minimizing data-driven computing is an emerging field of computational mechanics, which reformulates the classical boundary-value problem as a discrete-continuous optimization problem, seeking a minimum distance between the discrete material data space and the physical constraint space. However, the optimization problem becomes non-convex due to the limit equilibrium instability in snap-through phenomenon, which makes it challenging to obtain a desired convergent solution, especially near critical points. Towards this end, we propose a data-driven computational framework by virtue of the structural stability theory for analyzing snap-through problems. First, an auxiliary linearized perturbed system is established at each equilibrium state (i.e., convergent data-driven solution), which can be solved with the stiffness-based method and data-based method. Then, a stability indicator is employed to detect the critical point and the corresponding buckling mode, which is utilized to construct a proper start point to trace the equilibrium path in the neighborhood of critical points. Several numerical examples are performed to evaluate the reliability of the proposed framework. It is proved that even for the complex problem with nonlinear constitutive behavior, the proposed framework can correctly trace the entire equilibrium path of snap-through behavior.

Learning Data for Neural-Network-Based Numerical Solution of PDEs: Application to Dirichlet-to-Neumann Problems

We propose neural-network-based algorithms for the numerical solution of boundary-value problems for the Laplace equation. Such a numerical solution is inherently mesh-free, and in the approximation process, stochastic algorithms are employed. The chief challenge in the solution framework is to generate appropriate learning data in the absence of the solution. Our main idea was to use fundamental solutions for this purpose and make a link with the so-called method of fundamental solutions. In this way, beyond the classical boundary-value problems, Dirichlet-to-Neumann operators can also be approximated. This problem was investigated in detail. Moreover, for this complex problem, low-rank approximations were constructed. Such efficient solution algorithms can serve as a basis for computational electrical impedance tomography.

Weak Solution for a Fractional Langevin Inclusion with the Katugampola–Caputo Fractional Derivative

In this work, we examine the existence of weak solution for a class of boundary value problems involving fractional Langevin inclusion with the Katugampola–Caputo fractional derivative under specified conditions contain the Pettis integrability assumption. The Mönch fixed point theorem is used with the weak noncompactness measure approach to investigate the existence results. In order to illustrate our results, we present an example.

Integro-Differential Boundary Conditions to the Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations

In this paper, we introduce and study a new class of boundary value problems, consisting of a mixed-type ψ1-Hilfer and ψ2-Caputo fractional order differential equation supplemented with integro-differential nonlocal boundary conditions. The uniqueness of solutions is achieved via the Banach contraction principle, while the existence of results is established by using the Leray–Schauder nonlinear alternative. Numerical examples are constructed illustrating the obtained results.

A class of moving boundary problems with a source term: application of a reciprocal transformation

We consider a new Stefan-type problem for the classical heat equation with a latent heat and phase-change temperature depending of the variable time. We prove the equivalence of this Stefan problem with a class of boundary value problems for the nonlinear canonical evolution equation involving a source term with two free boundaries. This equivalence is obtained by applying a reduction to a Burgers equation and a reciprocal-type transformations. Moreover, for a particular case, we obtain a unique explicit solution for the two different problems.

A nonlinear Dirichlet problem involving terms with logarithmic growth

We study existence and regularity of weak solutions for a class of boundary value problems, whose form is − div ( log ( 1 + | ∇ u | ) | ∇ u | m ( x ) ∇ u ) + u | ∇ u | log ( 1 + | ∇ u | ) = f ( x ) , in Ω u = 0 , on ∂ Ω where both the principal part and the lower order term have a logarithmic growth with respect to the gradient of the solutions. We prove that the solutions, due to the regularizing effect given by the lower order term, belong to the Orlicz–Sobolev space generated by the function s log ( 1 + | s | ) even for L 1 ( Ω ) data.