Cauchy Problem Research Articles

Nonlocal separable elliptic and parabolic equations and applications

The regularity properties of nonlocal anisotropic elliptic equations with parameters are investigated in abstract weighted Lp spaces. The equations include the variable coefficients and abstract operator function A = A (x) in a Banach space E in leading part. We find the sufficient growth assumptions on A and appropriate symbol polynomial functions that guarantee the uniformly separability of the linear problem. It is proved that the corresponding anisotropic elliptic operator is sectorial and is also the negative generator of an analytic semigroup. Byusing these results, the existence and uniqueness of maximal regular solution of the nonlinear nonlocal anisotropic elliptic equation is obtained in weighted Lp spaces. In application, the maximal regularity properties of the Cauchy problem for degenerate abstract anisotropic parabolic equation in mixed Lp norms, the boundary value problem for anisotropic elliptic convolution equation, the Wentzel-Robin type boundary value problem for degenerate integro-differential equation and infinite systems of degenerate elliptic integro-differential equations are obtained.

Cauchy problem for the biharmonic equation in an unbounded region

The article studies the continuation of the solution of the Cauchy problem for the biharmonic equation in the domain G from its known values on the smooth part S of the boundary дG. The considered problem belongs to the problems of mathematical physics, in which there is no continuous dependence of solutions on the initial data. It is assumed that the solution to the problem exists and is continuously differentiable in a closed domain with exactly given Cauchy data. For this case, an explicit formula for the continuation of the solution is established.

Linear stabilization for a degenerate wave equation in non divergence form with drift

We consider a degenerate wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.

Mixed C-Cosine of Continuous Linear Operators on Complex p-Banach Spaces

In the present work, we define and examine the mixed C-cosine families of continuous linear operators on complex p-Banach spaces where 0 p 1. Finally, we will demonstrate several results on the mixed C-cosine families of continuous linear operators on complex p-Banach spaces where 0 p 1. Finally, we give an application related to the second order abstract Cauchy problem.

Исследование задачи оптимального нагрева жидкого продукта жидким теплоносителем в емкости с рубашкой

Two problems of optimal control of heating of a liquid product in a heating tank with a liquid coolant have been posed and solved on the Pontryagin maximum principle. In the first task, the problem of the speed of heating the product to a given temperature is solved. In the second task, we are looking for a control in which the product is heated for a given time to a given temperature with minimal coolant consumption. The description of the control object with a flow diagram, assumptions simplifying the compilation of the model, a mathematical model of the control object, formulations of optimal control problems and the solutions obtained are given. In the problem of heating speed, analytical expressions are obtained to determine the temperature of the product, the coolant and the control time. An example of solving the problem of heating speed is given with an illustration of the dynamic characteristics of the object variables and the optimal control found. In the problem of minimizing coolant consumption, the solution is obtained numerically. The problem of finding optimal trajectories of variables and control is formulated as a boundary value problem with missing initial conditions, the search for which is realized using the difference method of solving the boundary value problem. Based on the found initial conditions, the Cauchy problem of an extended system of equations is solved, and the optimal trajectory of changes in product temperature and coolant flow is obtained. An example of solving the problem of minimizing coolant consumption is given with a demonstration of the dynamic characteristics of the variables of the object and the optimal control found.

Relativistic electrons coupled with Newtonian nuclear dynamics

In the study of the electronic structure of heavy atoms, relativistic effects cannot be neglected and the Dirac operator naturally appears in place of the Schrödinger operator, raising a number of additional difficulties. The complexity of these systems has been addressed by various approximations.We consider a model consisting of differential equations coupling the time evolution of a finite number of relativistic electrons with the Newtonian dynamics of finitely many nuclei. The electrons are described by the Bogoliubov-Dirac-Fock (BDF) equation of quantum electrodynamics (QED). This system can be seen as a first step in the study of molecular dynamics phenomena, when both relativistic and quantum effects have to be taken into account for the electrons. We address the Cauchy problem for this model and prove global well-posedness.

On the Classical Fundamental Solution of the Cauchy Problem for One Class of Degenerate Parabolic Equations

On the Classical Fundamental Solution of the Cauchy Problem for One Class of Degenerate Parabolic Equations

Reliability Analysis and Numerical Simulation of the Five-Robot System with Early Warning Function

The rapid advancement of robotic technologies has demonstrated the significant potential of Multi-Robot Systems (MRS) for application across various fields, particularly in automation, manufacturing, and rescue operations. However, enhancing the reliability of Multi-Robot Systems, particularly in critical applications, has emerged as a primary focus of research. A mathematical model of a five-robot system, equipped with early warning capabilities, is developed using Markov process theory and the supplementary variable method in this paper. A model of an abstract Cauchy problem system is developed, employing semigroup theory to investigate the well-posedness of solutions for this five-robot system. The stability of the system is verified using analytical methods, confinal correlation theory, and modern functional analysis techniques. Several key reliability indicators are presented using the eigenvector method. Numerical simulations and comparative methods effectively demonstrate the efficacy of the proposed eigenvector method. Firstly, the innovation of this paper lies in the combination of qualitative and quantitative analyses to improve and enrich the theory and methods of repairable systems. Secondly, mathematical analysis methods and the mathematical software are employed to provide both analytical and numerical solutions for the system.

Critical mass for the Cauchy problem of a chemotaxis model with indirect signal production mechanism

Critical mass for the Cauchy problem of a chemotaxis model with indirect signal production mechanism

The time-dependent harmonic oscillator revisited

Abstract We point out a rather effective approach for solving the time-dependent harmonic oscillator $\ddot q=-\omega^2 q$ under various regularity assumptions. Where $\omega(t)$ is $C^1$ this is reduced to Hamilton equation for the angle variable $\psi$ alone (the action variable $I$ is obtained by quadrature). The fixed point theorem for the integral equation equivalent to the generic Cauchy problem for $\psi(t)$ 
yields a sequence $\{\psi^{(h)}\}_{h\in \mathbb{N}_0}$ converging to $\psi$ rather fast; if $\omega$ varies slowly or little, already $\psi^{(0)}$ approximates $\psi$ well for rather long time lapses. The discontinuities of $\omega$, if any, determine those of $\psi,I$. 
The zeroes of $q,\dot q$ are investigated via Riccati equations. Our approach may simplify the study of: upper and lower bounds 
on the solutions; the stability of the trivial one; parametric resonance when $\omega(t)$ is periodic; the adiabatic invariance of $I$; asymptotic expansions in a slow time parameter $\varepsilon$; time-dependent driven and damped parametric oscillators; etc.

Schottky-Invariant p-Adic Diffusion Operators

A parametrised diffusion operator on the regular domainΩ of a p-adic Schottky group is constructed. It is defined as an integral operator on the complex-valued functions on Ω which are invariant under the Schottky group Γ, where integration is against the measure defined by an invariant regular differential 1-form ω. It is proven that the space of Schottky invariant L2-functions on Ω outside the zeros of ω has an orthonormal basis consisting of Γ-invariant extensions of Kozyrev wavelets which are eigenfunctions of the operator. The eigenvalues are calculated, and it is shown that the heat equation for this operator provides a unique solution for its Cauchy problem with Schottky-invariant continuous initial conditions supported outside the zero set of ω, and gives rise to a strong Markov process on the corresponding orbit space for the Schottky group whose paths are càdlàg.

Well-posedness for strongly damped abstract Cauchy problems of fractional order

Let X be a complex Banach space and B be a closed linear operator with domain $\mathcal{D}(B) \subset X,\,\, a,b,c,d\in\mathbb{R},$ and $0 \lt \beta \lt \alpha.$ We prove that the problem \begin{equation*} u(t) -(aB+bI)(g_{\alpha-\beta}\ast u)(t) -(cB+dI)(g_{\alpha}\ast u)(t) = h(t), \quad t\geq 0, \end{equation*} where $g_{\alpha}(t)=t^{\alpha-1}/\Gamma(\alpha)$ and $h:\mathbb{R}_+\to X$ is given, has a unique solution for any initial condition on $\mathcal{D}(B)\times X$ as long as the operator B generates an ad-hoc Laplace transformable and strongly continuous solution family $\{R_{\alpha,\beta}(t)\}_{t\geq 0} \subset \mathcal{L}(X).$ It is shown that such a solution family exists whenever the pair $(\alpha,\beta)$ belongs to a subset of the set $(1,2]\times(0,1]$ and B is the generator of a cosine family or a C0-semigroup in $X.$ In any case, it also depends on certain compatibility conditions on the real parameters $a,b,c,d$ that must be satisfied.

Solution of Cauchy and Boundary Problems of Discrete Multiplicative-Poverative-Additive Derivative Third-Order Equation

Solution of Cauchy and Boundary Problems of Discrete Multiplicative-Poverative-Additive Derivative Third-Order Equation

Solutions of Cauchy Problems for the Gardner Equation in Three Spatial Dimensions

In this paper, we generalize the 2 + 1-dimensional Gardner (2DG) equation to three spatial dimensions, i.e., 3 + 1 and 3 + 2 dimensions, and construct the solutions of the Cauchy problems and Lax pairs for the Gardner equation in three spatial dimensions via a novel non-local d-bar formalism. Several new long derivative operators Dx, Dy and Dt are introduced to study the initial value problems for the Gardner equation in three spatial dimensions. It follows that Propositions 1 and 3 summarize the main results of this paper.

ЗАДАЧА ПРОДОВЖЕННЯ ПЛОСКОМЕРИДІАННОГО МАГНІТОСТАТИЧНОГО ПОЛЯ З ПЛОСКОЇ ГРАНИЧНОЇ ПОВЕРХНІ ФЕРОГМАГНЕТИКА

The problem of axisymmetric steady magnetic field continuation from flat boundary surface of ferromagnetic is formulated for magnetic flux and scalar potential. Boundary conditions for magnetic flux are not classical because on boundary magnetic flux is unknown and its normal derivative equal zero. The formulation for scalar potential is Cauchy's problem for elliptical partial differential equation. Analytical solutions of the problem are obtained by method of partial solutions, which depend on parameter continuously, and Hankel's integral transformation. It is shown that there are similar properties in problems of axisymmetric fields continuation from flat boundaries of ideal conductor for magnetic field and conductor for steady electric field. It is fixed that field lines, which bound unknown profile, are determined directly by solution of Cauchy's problem for one from two functions. Equipotential lines are calculated to determine of electromagnetic pole profile. References 7, figures 2, table 1.

On the abstract Cauchy problem for evolution equations in a complete random normed module

In this paper, we first study the abstract Cauchy problem for evolution equations in a complete random normed module, and further some basic results of the abstract Cauchy problem derived from an almost surely locally bounded C0–semigroup and its infinitesimal generator are presented. Meantime, the example also exhibits the necessity of the almost surely local boundedness for such a C0–semigroup. Then, based on the above work, a characterization for an evolution equation in a complete random normed module to be well-posed is established, which improves and generalizes some known results.

The Cauchy problem for a compressible generic two-fluid model with large initial data: Global existence and uniqueness

The Cauchy problem for a compressible generic two-fluid model with large initial data: Global existence and uniqueness

On the Numerical Approximation and Optimization Techniques for Solving an Inverse Cauchy Problem of Viscous-Burgers’ Equation

This paper deals with some inverse problems for nonlinear time-dependent PDEs in one spatial dimension, we investigate an inverse Cauchy problem that is settled by the nonlinear viscous Burgers equation. The viscous Burgers equation is a partial differential equation that is encountered in fluid dynamics studies, particularly in the domain of upward flow. The simplified model of the viscous Burgers equation explains the behavior of incompressible viscous fluid. The inverse Burgers problem belongs to a class of problems called ill-posed problems, which implies that there may be multiple sets of initial andor boundary conditions that result in the same solution of the Burgers equation. To obtain robust and reliable solutions, it is essential to use regularization and cross-validation methods. However, it is often difficult to solve analytically, so numerical approaches are developed to overcome this difficulty. Domain decomposition (DDM) was used with alternative iterative methods. We performed a numerical reconstruction of the velocity and normal stress tensor that were vanished on an inaccessible part of the boundary using the over-prescribed noisy data obtained on the other accessible part of the boundary.

Solving a system of Caputo-Hadamard fractional differential equations via Perov’s fixed point theorem

Abstract In this study, we discuss the existence and the uniqueness of the solution to Caputo-Hadamard Cauchy problems for a system of fractional differential equations, by using Perov’s fixed point theorem. Finally, two examples are provided to illustrate our results.

Nonclassical dynamical behavior of solutions of partial differential-difference equations

<p>For partial differential-difference equations, a review of results regarding the relation between the type of the equation and dynamical properties of its solutions is provided. This includes the case of elliptic equations with timelike independent variables: Their solutions acquire dynamical properties (more exactly, behave as solutions of parabolic equations). The following approach to classify differential-difference equations into types, based on the property of differential-difference operators to be Fourier multipliers is applied in the following manner: An operator is treated to be elliptic if the real part of its symbol is positive, while the parabolic and hyperbolic types are defined correspondingly. It is shown that the proposed approach (being a natural extension of the classical notion of the ellipticity) is reasonable. On the other hand, fundamental novelties (compared with the classical theory of partial differential equations) occur as well. We provide conditions which guarantee the following results. For the half-space (half-plane) Dirichlet problem for elliptic equations, integral representations (of the Poisson type) of solutions are constructed, which are infinitely differentiable outside the boundary hyperplane (plane), and the asymptotic closeness of solutions (as the timelike independent variable unboundedly increases) takes place. For the Cauchy problem for parabolic equations, the same is valid, but we deal with the classical time instead of the timelike independent variable. For hyperbolic equations, multiparameter families of infinitely smooth global solutions are constructed. The said (sufficient) conditions restrict the sign of the real part of the symbol for the differential-difference operator with respect to spatial or spacelike independent variables. In a number of special cases, they might be weakened such that symbols with sign-changing real parts are admitted. The objective of the study is to observe the current stage of the classification issues for partial differential-difference equations in terms of the aforementioned approach: The ellipticity of a Fourier multiplier is defined by means of the sign (of there real part) of its symbol. Since both differential and translation operators are Fourier multipliers, methods of Fourier analysis are applicable in this study: We apply the Fourier transformation to the original partial differential-difference equation, solve the obtained ordinary differential equation, and apply the inverse Fourier transformation to the obtained solution. The main contribution obtained within this study is an efficient (workable) type concept for the fundamentally new extension of the class of partial differential equations, which is the class of partial differential-difference equations.</p>