Well-posedness Of Boundary Value Problem Research Articles

Existence and uniqueness of well-posed fractional boundary value problem.

In this paper, the existence and uniqueness of solution for a fractional differential model involving well-posed boundary conditions and implicit fractional differential equation is considered. The desired goals are achieved by using Banach contraction principle and Scheafer's fixed point theorem. To show the results applicability some examples are presented. The basic mathematical concept of well-posed fractional boundary value issues is investigated in this study. It dives into the existence and uniqueness of these difficulties, offering light on the conditions that allow for both practical and singular solutions. This study contributes to a better knowledge of fractional calculus and its applications in a variety of scientific and technical areas, giving significant insights for both scholars and practitioners.

General Solutions for MHD Motions of Ordinary and Fractional Maxwell Fluids through Porous Medium When Differential Expressions of Shear Stress Are Prescribed on Boundary

Some MHD unidirectional motions of the electrically conducting incompressible Maxwell fluids between infinite horizontal parallel plates incorporated in a porous medium are analytically and graphically investigated when differential expressions of the non-trivial shear stress are prescribed on the boundary. Such boundary conditions are usually necessary in order to formulate well-posed boundary value problems for motions of rate-type fluids. General closed-form expressions are established for the dimensionless fluid velocity, the corresponding shear stress, and Darcy’s resistance. For completion, as well as for comparison, all results are extended to a fractional model of Maxwell fluids in which the time fractional Caputo derivative is used. It is proven for the first time that a large class of unsteady motions of the fractional incompressible Maxwell fluids becomes steady in time. For illustration, three particular motions are considered, and the correctness of the results is graphically proven. They correspond to constant or oscillatory values of the differential expression of shear stress on the boundary. In the first case, the required time to reach the steady state is graphically determined. This time declines for increasing values of the fractional parameter. Consequently, the steady state is reached earlier for motions of the ordinary fluids in comparison with the fractional ones. Finally, the fluid velocity, shear stress, and Darcy’s resistance are graphically represented and discussed for the fractional model.

A problem with periodic boundary conditions for the non-Fourier heat equation

All diffusion equations are based on the infinite velocity of potential fields, which leads to well-known paradoxes. Consequently, in non-stationary processes, the evolution of these quantities do not completely obey the above equations due to the lack of parameters in them that take into account the finite rate of potential growth.In the heat conduction theory, numerous generalizations of the Fourier law are used as a remedy for these issues. The article gives a brief overview of generalizations of the Fourier law. Some mathematical issues of well-posed boundary value problems for the Guyer-Krumhansl model are discussed. As an application, a boundary value problem for a general quasilinear equation with periodic boundary conditions is considered. Schauder-type a priori estimates are established and the uniqueness of the solution is proved.

Geometric optics expansion for weakly well-posed hyperbolic boundary value problem: The glancing degeneracy

This article aims to finalize the classification of weakly well-posed hyperbolic boundary value problems in the half-space. Such problems with loss of derivatives are rather classical in the literature and appear for example in (Arch. Rational Mech. Anal. 101 (1988) 261–292) or (In Analyse Mathématique et Applications (1988) 319–356 Gauthier-Villars). It is known that depending on the kind of the area of the boundary of the frequency space on which the uniform Kreiss–Lopatinskii condition degenerates then the energy estimate can include different losses. The three first possible areas of degeneracy have been studied in (Annales de l’Institut Fourier 60 (2010) 2183–2233) and (Differential Integral Equations 27 (2014) 531–562) by the use of geometric optics expansions. In this article we use the same kind of tools in order to deal with the last remaining case, namely a degeneracy in the glancing area. In comparison to the first cases studied we will see that the equation giving the amplitude of the leading order term in the expansion, and thus initializing the whole construction of the expansion, is not a transport equation anymore but it is given by some Fourier multiplier. This multiplier needs to be invert in order to recover the first amplitude. As an application we discuss the existing estimates of (Discrete Contin. Dyn. Syst., Ser. B 23 (2018) 1347–1361; SIAM J. Math. Anal. 44 (2012) 1925–1949) for the wave equation with Neumann boundary condition.

Well-posed problems for the Laplace-Beltrami operator on a punctured two-dimensional sphere

An arbitrary point is removed from a three-dimensional Euclidean space on a two-dimensional sphere. The new well-posed solvable boundary value problems for the corresponding Laplace-Beltrami operator on the resulting punctured sphere are presented. To formulate the well-posed problems some properties of Green's function of the Laplace-Beltrami operator on a two-dimensional sphere are previously studied in detail.

New trends in couple-stress hyperelasticity

This communication reconsiders and reorganises parts of existing knowledge that refers to Cosserat-type hyperelasticity formulations in a manner that respects and pays tribute to the early development pattern of the couple-stress theory and, also, associates with it more recent findings and relevant discoveries related to a certain type of polar hyperelastic behaviour of fibrous composites. It further shows that these different branches of couple-stress theory can both emanate, as special cases, from a common, more general, advanced theoretical hyperelasticity framework. It thus reveals that proper completion of any relevant theoretical formulation requires in advance specification of a pair of kinematic parameters, each of which (1) is related, in a virtual or actual manner, to the observed global, macroscopic elastic deformation of the material, but (2) is principally relevant to the polar part of the observed material response. The first of these parameters represents a pseudovector field whose gradient is energetically reciprocal to the emerging couple-stress field, while the second represents either a vector or a pseudovector field that serves specific constitutional needs characterising the source of the anticipated polar material response. Different versions of couple-stress theory formulations, thus, are obtained by appropriately choosing or suitably tuning that pair of kinematic parameters. It is also seen that, regardless of the employed couple-stress theoretical model, full solution of a relevant well-posed boundary value problem is generally achievable with use of a two-step solution process. The first step includes determination of the deviatoric couple-stress and the actual spin vector of the global material deformation. In the second step, these initial findings enable formation of an additional differential equation whose solution leads to determination of the spherical part of the couple-stress.

Comparative Study of Guided Electromagnetic Wave Propagation for Two Models of an Open Tape Helix

Rigorous methods to solve the homogeneous boundary value problems and the techniques to resolve illposed mathematical issues arising in the problem formulation of electromagnetic wave propagation of an Open tape helix slow wave structure with anisotropically conducting and completely conducting model is presented. The dispersion equation of both the models require convergent coeefficients of the infinte linear homogeneous equations whose determinant of the coefficent matrix is zero. The coefficient matrix of the anisotropically conducting model is rapidly convergent, while in the case of completely conducting model, the matrix entries which are infinite summation series representations turns out to be divergent, the not so well-posed boundary value problem is regularised with the method of mollification functions. The dispersion characteristics are plotted after truncating the infinite series expansions to adequate number of terms and summing the terms of converging series expansions(after regularization in the case of the completely conducting tape-helix model) in dispersion equations. The tape-current distribution for both models are estimated from the null-space vector of the truncated coefficient matrix corresponding to a particular (β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> – <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ) root of the dispersion equation. A comparison of the numerical results for both models show that the neglect of the perpendicular tape-current density component entails a substantial modification of the dispersion characteristics of guided waves supported by an open tape helix.

A new method for the evaluation of the vacuum boundary in circular and D-shaped Tokamaks

A new method for the evaluation of the vacuum boundary in a Tokamak is introduced. It is a two-step iterative algorithm that converges to the actual value of the boundary location only after a few iterations. The key point in this method is to solve a well-posed boundary value problem that satisfies a set of exact boundary conditions. A Cauchy problem in steady heat-conduction problem is used to explain the method first. Then, inverse evaluation of the vacuum boundary location in circular as well as D-shaped Tokamaks are considered. For circular Tokamak, the method does not require any regularization. The algorithm is quite simple and very effective. Numerical examples are presented to study the performance of the algorithm in the presence of noise.

Gaussian process regression constrained by boundary value problems

We develop a framework for Gaussian processes regression constrained by boundary value problems. The framework may be applied to infer the solution of a well-posed boundary value problem with a known second-order differential operator and boundary conditions, but for which only scattered observations of the source term are available. Scattered observations of the solution may also be used in the regression. The framework combines co-kriging with the linear transformation of a Gaussian process together with the use of kernels given by spectral expansions in eigenfunctions of the boundary value problem. Thus, it benefits from a reduced-rank property of covariance matrices. We demonstrate that the resulting framework yields more accurate and stable solution inference as compared to physics-informed Gaussian process regression without boundary condition constraints.

WKB expansions for weakly well-posed hyperbolic boundary value problems in a strip: Time depending loss of derivatives

We are interested in geometric optics expansions for linear hyperbolic systems of equations defined in the strip [Formula: see text]. More precisely the aim of this paper is to describe the influence of the boundary conditions on the behavior of the solution. This question has already been addressed in [A. Benoit, Wkb expansions for hyperbolic boundary value problems in a strip: Selfinteraction meets strong well-posedness, J. Inst. Math. Jussieu 19(5) (2020) 1629–1675] for stable boundary conditions. Here we do not require that the boundary conditions lead to strongly well-posed problems but only to weakly well-posed problems (that is loss(es) of derivatives are possible). The question is thus to determine what can be the minimal loss of derivatives in the energy estimate of the solution. The main result of this paper is to show, thanks to geometric optics expansions, that if the strip problem admits a boundary in the so-called [Formula: see text]-class of [S. Benzoni-Gavage, F. Rousset, D. Serre and K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A 132(5) (2002) 1073–1104] then the loss of derivatives shall be at least increasing with the time of resolution. More precisely this loss is bounded by below by a step function increasing with respect to time which depends on the minimal time needed to perform a full regeneration of the wave packet.

Mathematical Simulations of Deformation for the Rotation Shells with Variable Wall Thickness

Well-posed boundary value problems are constructed for calculating rotation shells of with a stiffness variable along the meridian in two directions, and also with variable bilateral with respect to the reference surface with the shell wall thickness. Algorithms for the numerical integration of systems of differential equations with variable coefficients are discussed.

A deep artificial neural network architecture for mesh free solutions of nonlinear boundary value problems

Seeking efficient solutions to nonlinear boundary value problems is a crucial challenge in the mathematical modelling of many physical phenomena. A well-known example of this is solving the Biharmonic equation relating to numerous problems in fluid and solid mechanics. One must note that, in general, it is challenging to solve such boundary value problems due to the higher-order partial derivatives in the differential operators. An artificial neural network is thought to be an intelligent system that learns by example. Therefore, a well-posed mathematical problem can be solved using such a system. This paper describes a mesh free method based on a suitably crafted deep neural network architecture to solve a class of well-posed nonlinear boundary value problems. We show how a suitable deep neural network architecture can be constructed and trained to satisfy the associated differential operators and the boundary conditions of the nonlinear problem. To show the accuracy of our method, we have tested the solutions arising from our method against known solutions of selected boundary value problems, e.g., comparison of the solution of Biharmonic equation arising from our convolutional neural network subject to the chosen boundary conditions with the corresponding analytical/numerical solutions. Furthermore, we demonstrate the accuracy, efficiency, and applicability of our method by solving the well known thin plate problem and the Navier-Stokes equation.

Complexes from Complexes

This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a systematic procedure which, starting from well-understood differential complexes such as the de Rham complex, derives new complexes and deduces the properties of the new complexes from the old. We relate the cohomology of the output complex to that of the input complexes and show that the new complex has closed ranges, and, consequently, satisfies a Hodge decomposition, Poincar\'e type inequalities, well-posed Hodge-Laplacian boundary value problems, regular decomposition, and compactness properties on general Lipschitz domains.

GREEN'S FUNCTIONS AND CORRECT RESTRICTIONS OF THE POLYHARMONIC OPERATOR

In this paper, for completeness of presentation, we give explicitly the Green's functions for the classical problems - Dirichlet, Neumann, and Robin for the Poisson equation in a multidimensional unit ball. There are various ways of constructing the Green's function of the Dirichlet problem for the Poisson equation. For many types of areas, it is built explicitly. Recently, there has been renewed interest in the explicit construction of Green's functions for classical problems. The Green's function of the Dirichlet problem for a polyharmonic equation in a multidimensional ball is constructed in an explicit form, and for the Neumann problem the construction of the Green's function remains an open problem. The paper gives a constructive way of constructing the Green's function of Dirichlet problems for a polyharmonic equation in a multidimensional ball. Finding general well-posed boundary value problems for differential equations is always an urgent problem. In this paper, we briefly outline the theory of restriction and extension of operators and describe well-posed boundary value problems for a polyharmonic operator.

Non-Iterative Solution Methods for Cauchy Problems for Laplace and Helmholtz Equation in Annulus Domain

This note is concerned with two new methods for the solution of a Cauchy problem. The first method is based on homotopy-perturbation approach which leads to solving a series of well-posed boundary value problems. No regularization is needed in this method. Laplace and Helmholtz equations are considered in an annular region. It is also proved that the homotopy solution for the Laplace operator converges to the actual exact solution. The second method is also non-iterative. It is based on the application of the Green’s second identity which leads to a moment problem for the unknown boundary condition. Tikhonov regularization is used to obtain a stable and close approximation of the missing boundary condition. A number of examples are used to study the applicability of the methods with the presence of noise.

Explicit characteristic equations for integral operators arising from well-posed boundary value problems of finite beam deflection on elastic foundation

&lt;abstract&gt;&lt;p&gt;Characteristic equations for the whole class of integral operators arising from arbitrary well-posed two-point boundary value problems of finite beam deflection resting on elastic foundation are obtained in terms of $ 4 \times 4 $ matrices in block-diagonal form with explicit $ 2 \times 2 $ blocks.&lt;/p&gt;&lt;/abstract&gt;

Application of the Method of Non-Classical Approach to Certain Problems of the Theory of Anisotropic Space Plazma in Fluid Description

In this paper, the new well-posed boundary value problem to the system of singular nonlinear differential equations is considered, which describes the stationary radial outflow of anisotropic plasma from the Sun (solar wind). These equations are obtained on the basis of 16-moment MHD (magnetohydrodynamic) transport equations for a collisionless magnetized plasma, which takes into account the temperature anisotropy relative to the direction of the magnetic field and the heat flux carried by the wind. This is a generalization of the classical isotropic Parker model taking into account the effects of anisotropy. In this paper, the equations under study are characterized as a non-autonomous nonlinear system of ordinary differential equations the coefficients in which degenerate and simultaneously have singularities. These equations are related to an unsolved problem in the general theory of ODEs (Ordinary Differential Equations). At first, according to the conditions of the coefficients of the equations, a non-classical boundary value problem is set, and the solvability is established for the same non-autonomous and nonlinear system of equations under consideration. The found analytical solution reconstructs numerical solutions, which are simultaneously automatically established by classical formulation of boundary value problem. Parker’s solutions are also partially included in this obtained class of solutions, which is presented with strictly proves. Further, by means of the methods of “ -regularization” and “fixed point” the theorem of solvability for the considered differential equations is obtained. After constructed nonsingular system equations with well-posed boundary value problem, the analytical solutions are founded. Using the sketch of graph of these solutions their family is established.

О разрешимости обратных задач восстановления параметров в эллиптических уравнениях

Изучается разрешимость обратных задач нахождения вместе с решением $u(x, t)$ также положительного параметра $\alpha$ в дифференциальных уравнениях $$ u_{tt}+\alpha\Delta u-\beta u=f(x,t),\;\alpha u_{tt}+\Delta u-\beta u=f(x,t) $$ ($t\in(0,T)$, $x=(x_1,\dots,x_n)\in\Omega\subset\mathbb{R}^n$, $\Delta$ – оператор Лапласа по переменным $x_1,\dots,x_n$). В качестве дополнения к краевым условиям, определяющим корректную краевую задачу для эллиптических уравнений, используется условие линейного финально-интегрального переопределения. Доказываются теоремы существования и единственности регулярных (имеющих все обобщенные по С. Л. Соболеву производные, входящие в уравнение) решений. We study solvability of the inverse problems of finding, alongside the solution $u(x, t)$, the positive parameter $\alpha$ in the differential equations $$ u_{tt}+\alpha\Delta u-\beta u=f(x,t),\;\alpha u_{tt}+\Delta u-\beta u=f(x,t) $$ Where $t\in(0,T)$, $x=(x_1,\dots,x_n)\in\Omega\subset\mathbb{R}^n$, and $\Delta$ – the Laplace operator in variables $x_1,\dots,x_n$. As a complement to the boundary conditions defining a well-posed boundary value problem for elliptic equations, we use the conditions of the linear final integral overdetermination. We prove the existence and uniqueness theorems for regular solutions, those having all generalized in the S. L. Sobolev sense derivatives in the equation.

Second law thermodynamic analysis of thermo-magnetic Jeffery–Hamel dissipative radiative hybrid nanofluid slip flow: existence of multiple solutions

In this article, a detailed theoretical examination is conducted for the steady, incompressible, MHD hybrid nanofluid (Al2O3/Cu–water) dissipative slip flow between non-parallel (i.e., diverging or converging) shrinking/stretching walls in the presence of appreciable thermal radiation. Appropriate similarity transformations have been applied to render the hybrid transport model conservation equations dimensionless. The diffusion flux approximation is employed for radiative heat transfer. The well-posed boundary value problem has been analyzed for solution bifurcations with a predictor homotopy analysis method (PHAM) along with stability analysis. The eigenvalues obtained predict the upper branch (Ist solution) to be physically acceptable. The critical values ( $$\chi_{c} \le \chi < 0$$ ) of stretching/shrinking parameter are evaluated by varying slip and magnetic body force parameters. The impact of significant parameters on skin friction coefficient, Nusselt number and entropy generation number is also visualized and elaborated in detail. With stronger magnetic field parameter (M), i.e., enhancement in magnetohydrodynamic Lorentz drag force, the existence domain of the dual solutions is shown to be expanded and the critical points of the stretching parameter ( $$\chi$$ ) for $$M = 0.1,0.5,1$$ are identified, respectively, as $$\chi_{c} = - 3.2455, - 3.2680,$$ $$- 3.2987$$ . The upper branch of skin friction, Cfr decreases as the volume fraction $$\varphi$$ increases whereas the lower branch increases with an increase in the volume fraction of copper nanoparticles. Entropy generation is observed to be elevated with stronger radiation, inertial effect (Reynolds number) and Prandtl number (i.e., lower thermal conductivity) whereas it is suppressed with increasing hybrid nanoparticle volume fraction and wall hydrodynamic slip effect. Magnetic field is found to induce a weak modification in entropy generation. Increasing angular coordinate $$\left( {\xi = \frac{\theta }{\beta }} \right)$$ is observed to elevate the entropy generation. The computations find applications in thermo-magnetic nozzle design, electromagnetic propulsion systems and electroconductive materials processing.

Exact Green's formula for the fractional Laplacian and perturbations

Let Ω be an open, smooth, bounded subset of $ \mathbb{R}^n $. In connection with the fractional Laplacian $(-\Delta )^a$ ($a&gt;0$), and more generally for a $2a$-order classical pseudodifferential operator (ψdo) $P$ with even symbol, one can define the Dirichlet value $\gamma _0^{a-1}u$, resp. Neumann value $\gamma _1^{a-1}u$ of $u(x)$, as the trace, resp. normal derivative, of $u/d^{a-1}$ on $\partial \Omega $, where $d(x)$ is the distance from $x\in \Omega $ to $\partial \Omega $; they define well-posed boundary value problems for $P$.&#x0D; A Green's formula was shown in a preceding paper, containing a generally nonlocal term $(B\gamma _0^{a-1}u,\gamma _0^{a-1}v)_{\partial \Omega }$, where $B$ is a first-order ψdo on $\partial \Omega $. Presently, we determine $B$ from $L$ in the case $P=L^a$, where $L$ is a strongly elliptic second-order differential operator. A particular result is that $B=0$ when $L=-\Delta $, and that $B$ is multiplication by a function (is local) when $L$ equals $-\Delta $ plus a first-order term. In cases of more general $L$, $B$ can be nonlocal.