Unique Solvability Research Articles

Coefficient inverse problem for an equation of mixed parabolic-hyperbolic type with a non-characteristic line of type change

In this paper, we study the direct and two inverse problems for a model equation of mixed parabolic-hyperbolic type. In the direct problem, the Tricomi problem for this equation with a non-characteristic line of type change is considered. The unknown of the inverse problem is the variable coefficient at the lowest derivative in the parabolic equation. To determine it, two inverse problems are studied: with respect to the solution defined in the parabolic part of the domain, the integral overdetermination condition (inverse problem 1) and one simple observation at a fixed point (inverse problem 2) are given. Theorems on the unique solvability of the formulated problems in the sense of classical solution are proved.

BOUNDARY-DOMAIN INTEGRAL EQUATIONS FOR VARIABLE-COEFFICIENT HELMHOLTZ BVPs IN 2D

In this paper, we construct boundary-domain integral equations (BDIEs) of the Dirichlet and mixed boundary value problems for a two-dimensional variable-coefficient Helmholtz equation. Using an appropriate parametrix, these problems are reduced to several BDIE systems. It is shown that the BVPs and the formulated BDIE systems are equivalent. Fredholm properties and unique solvability and invertibility of BDIE systems are investigated in appropriate Sobolev spaces.

About one approach for solving bounary-value problems of differential equation of pantograph type

Pantograph equations have been studied for a long time. In 1940, K. Mahler introduced functional differential equations of this type into number theory. In 1971, J. Ockendon used a functional differential equation with a transformed argument, , to describe the dynamics of the pantograph of an electric locomotive. Subsequently, Pantograph-type equations were studied in the works of many authors. This article considers a boundary value problem for a differential equation of the pantograph type. To solve the posed boundary value problem, the parameterization method proposed by Professor D. Dzhumabaev is used. To do this, we denote the value of the function at the initial point of the segment under consideration by a parameter and perform a change of variable Then the solvability of the original boundary value problem is reduced to studying the solvability of the resulting Cauchy problem for the original equation and to a linear algebraic equation to determine the introduced parameter. Next, using the method of successive approximations, we find solutions to the Cauchy problem for the Pantograph type equation. The convergence of the resulting sequence and the convergence of its solution to the solution of the Cauchy problem of Pantograph type are proved. By requiring continuity of the free term, we establish its unique solvability. Substituting the resulting solution into a linear algebraic equation to determine the entered parameter, we calculate the values of the entered parameter using the original data. Substituting the resulting expressions into we find the solution to the original problem . And assuming the unique solvability of a linear algebraic equation, we establish the solvability of the boundary value problem for an equation of Pantograph type.

COMPOSITIONS OF FRACTIONAL DERIVATIVES AS DZHRBASHYAN - NERSESYAN DERIVATIVE

The issues of unique solvability for a special initial value problem to two classes of a linear inhomogeneous equation with a derivative of Dzhrbashyan - Nersesyan are investigated. In one of the classes, the operator at the unknown function is bounded, in the other it is sectorial. The representation of the composition of any number of Gerasimov - Caputo and/or Riemann - Liouville differential operators in the form of a Dzhrbashyan - Nersesyan derivative is also proved.

MIXED PROBLEM FOR A NONLINEAR IMPULSIVE DIFFERENTIAL EQUATION OF PARABOLIC TYPE

In this paper, we consider a nonlinear impulsive parabolic type partial differential equation with nonlinear impulsive conditions. Dirichlet type boundary value conditions with respect to spatial variable is used, and eigenvalues and eigenfunctions of the spectral problem are founded. The Fourier method of the separation of variables is applied. A countable system of nonlinear functional equations is obtained with respect to the Fourier coefficients of the unknown function. A theorem on a unique solvability of the countable system of nonlinear functional equations is proved by the method of successive approximations. A criteria of uniqueness and existence of a solution for the nonlinear impulsive mixed problem is obtained. A solution of the mixed problem is derived in the form of the Fourier series. The absolute and uniform convergence of the Fourier series is proved.

ON THE UNIQUE SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR DIFFERENTIAL EQUATIONS WITH PARAMETER

A linear boundary value problem for a differential equation with a parameter is investigated on a finite interval by the parameterization method. The studied boundary value problem with parameter is reduced to an equivalent multipoint boundary value problem with parameters by splitting the interval, introducing additional parameters at the points of splitting and new functions. The obtained equivalent boundary value problem contains Cauchy problems for ordinary differential equations with respect to new functions. By substituting the solution representation of the Cauchy problem into the boundary conditions and continuity conditions of the solution, a system of linear algebraic equations with respect to the introduced parameters is compiled. An algorithm for finding a solution to the boundary value problem with parameters is constructed. The formulation of the theorem on sufficient conditions of unique solvability of the boundary value problem with parameters is given. Sufficient conditions of its unique solvability are obtained in terms of the data of the original boundary value problem. An example showing the fulfillment of the conditions of the theorem is given.

An alternative potential method for mixed steady‐state elastic oscillation problems

We consider an alternative approach to investigate three‐dimensional exterior mixed boundary value problems (BVPs) for the steady‐state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, , has a compact boundary surface , which is divided into two disjoint parts, the Dirichlet part and the Neumann part , where the displacement vector (the Dirichlet‐type condition) and the stress vector (the Neumann‐type condition) are prescribed, respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed BVP in the form of a linear combination of the single‐layer and double‐layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov–Poincaré‐type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right‐hand sides of the resulting boundary integral equations system are vector functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate ‐based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector functions belonging to the Sobolev space and satisfying the Sommerfeld–Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the ‐based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses ‐Hölder continuity property in the closed domain with , where is an arbitrarily small number.

On the Boyarsky-Meyers estimate for the solution of the Dirichlet problem for a second-order linear elliptic equation with drift

We establish the increased integrability of the gradient of the solution to the Dirichlet problem for the Laplace operator with lower terms and prove the unique solvability of this problem.

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.

Second-order accurate and unconditionally stable algorithm with unique solvability for a phase-field model of 3D volume reconstruction

Three-dimensional (3D) volume reconstruction is an important technique in the fields of 3D printing, artificial limb, and medical diagnostic. This work aims to develop an unconditionally stable algorithm with desired accuracy and unique solvability for a phase-field model of 3D volume reconstruction. Based on scattered points of a target object, a smooth narrow volume is reconstructed by solving a Allen–Cahn-type equation with control function. Since the non-negative property of control function, the evolution of governing equation can associate with an energy dissipation law (i.e., energy stability). By modifying the nonlinear term in governing equation into an appropriate truncated form and utilizing a stabilized Crank–Nicolson-type discretization in time, we develop a linear, second-order accurate, and unconditionally energy-stable time-marching scheme to update the solution. The space is discretized with the finite difference method and the fully discrete energy stability is analytically estimated. We also prove the unique solvability of our proposed time-marching scheme. The accuracy, stability, and capability of reconstructing various 3D volumes are verified via extensive numerical simulations. To facilitate the interested readers, we provide the open source codes of 3D volume reconstruction of a teapot on corresponding author's personal website: https://cfdyang521.github.io.

Evolution Equations with Liouville Derivative on R without Initial Conditions

New classes of evolution differential equations with the Liouville derivative in Banach spaces are studied. Equations are considered on the whole real line and are not endowed by the initial conditions. Using the methods of the Fourier transform theory, we prove the unique solvability in the sense of classical solutions for the equation solved with respect to the Liouville fractional derivative with a bounded operator at the unknown function. This allows us to obtain the analogous result for the equation with a linear degenerate operator at the fractional derivative and with a spectrally bounded pair of operators. Abstract results are applied to obtain new results on the unique solvability of systems of ordinary differential equations, boundary problems to partial differential equations, and systems of equations.

Fully decoupled and high-order linearly implicit energy-preserving RK-GSAV methods for the coupled nonlinear wave equation

This paper is concerned with high-order accurate, linearly implicit and energy-preserving schemes for the coupled nonlinear wave equation. To this end, a novel auxiliary variable approach proposed in a recent paper (Ju et al., 2022) is applied to reformulate the governing equation in an equivalent system, which possesses a modified energy that consists of primary functional and quadratic functional. Subsequently, we utilize the extrapolation strategy/prediction–correction technique for treating the nonlinear terms of the equivalent system to obtain a linearized energy-preserving system. Then, a series of fully decoupled high-order linear energy-preserving schemes are constructed by using the symplectic Runge–Kutta (RK) methods. We show that the modified energy functional can be precisely conserved under certain circumstances for the coefficients of the symplectic RK methods, and the unique solvability of the proposed time-stepping scheme is analyzed. In numerical implementations, a class of linear algebraic equations with constant-coefficient matrices need to be solved at each time step and only less computational costs are required. Extensive numerical experiments of the coupled nonlinear wave equation with local/nonlocal diffusion operators are provided to illustrate the energy conservation law and the computational efficiency of the schemes in long-time simulations.

Symptotic behavior of solutions of the inhomogeneous Schrödinger equation on noncompact Riemannian manifolds

The paper studies the behavior of bounded solutions of the inhomogeneous Schrödinger equation on non-compact Riemannian manifolds under a variation of the right side of the equation. Various problems for homogeneous elliptic equations, in particular the Laplace-Beltrami equation and the stationary Schrödinger equation, have been considered by a number of Russian and foreign authors since the second half of the 20th century. In the first part of this paper, an approach to the formulation of boundary value problems based on the introduction of classes of equivalent functions will be developed. The relationship between the solvability of boundary value problems on an arbitrary non-compact Riemannian manifold with variation of inhomogeneity is also established. In the second part of the work, based on the results of the first part, properties of solutions of the inhomogeneous Schrödinger equation on quasi-model manifolds are investigated, and exact conditions for unique solvability of the Dirichlet problem and some other boundary value problems on these manifolds are found.

Forward–backward stochastic evolution equations in infinite dimensions and application to LQ optimal control problems

This paper focuses on the study of forward–backward stochastic evolution equations (FBSEEs), which are a class of nonlinear fully coupled forward–backward stochastic differential equations (FBSDEs), in infinite dimensions. Drawing inspiration from various linear-quadratic (LQ) optimal control problems, we apply a set of domination-monotonicity conditions that are more relaxed compared to general conditions. Within this framework, we employ the method of continuation to establish the unique solvability result and provide a pair of solution estimates. Notably, to address the challenges posed by the infinite-dimensional setting, we introduce a class of approximating equations, as Itô’s formula is not directly applicable. Conversely, these results find application in various LQ problems, where the stochastic Hamiltonian systems precisely correspond to the FBSEEs satisfying the aforementioned domination-monotonicity conditions. Consequently, by solving the corresponding stochastic Hamiltonian systems, we can obtain explicit expressions for the optimal controls.

Characterization of unique solvability of absolute value equations: an overview, extensions, and future directions

Characterization of unique solvability of absolute value equations: an overview, extensions, and future directions

Errata of the Unique Solvability of the Absolute Value Equation

Errata of the Unique Solvability of the Absolute Value Equation

Convergence Criteria for Fixed Point Problems and Differential Equations

We consider a Cauchy problem for differential equations in a Hilbert space X. The problem is stated in a time interval I, which can be finite or infinite. We use a fixed point argument for history-dependent operators to prove the unique solvability of the problem. Then, we establish convergence criteria for both a general fixed point problem and the corresponding Cauchy problem. These criteria provide the necessary and sufficient conditions on a sequence {un}, which guarantee its convergence to the solution of the corresponding problem, in the space of both continuous and continuously differentiable functions. We then specify our results in the study of a particular differential equation governed by two nonlinear operators. Finally, we provide an application in viscoelasticity and give a mechanical interpretation of the corresponding convergence result.

A linear, second-order accurate, positivity-preserving and unconditionally energy stable scheme for the Navier–Stokes–Poisson–Nernst–Planck system

Electrokinetic phenomena involving the transport of ions in fluids generally appears in biological system, hydrodynamics and geodynamics. The underlining mathematical models involve the nonlinear coupling between the fluid motion and ion transportation. Performing efficient, accurate and structure preserving computation for these electrohydrodynamic models is challenging due to their multiphysics and complex nature. The Navier–Stokes–Poisson–Nernst–Planck (NSPNP) system is a widely used electrokinetic model in the existing literature. In this paper, a linear, second-order accurate, positivity-preserving and unconditionally energy stable scheme is developed for the NSPNP system. First, a square root transformation is employed to ensure the positivity of ion concentrations. Next, a specific ordinary differential equation is introduced to deal with the nonlinear coupling terms satisfying the “zero-energy-contribution” feature, which is combined with the scalar auxiliary variable method for nonlinear potentials to achieve a decoupled energy stable scheme. Therefore, only several linearly independent equations need to be solved at each time step. The staggered grid finite difference method is used for the spatial discretizations, and the unconditional energy stability of the fully discrete scheme is established. Finally, the unique solvability of the scheme is proved. Numerical experiments are carried out to demonstrate the accuracy and performance of the proposed scheme, and the effect of electric Coulomb force on the flow and motion of ions are also illustrated.

Numerical simulation methods and analysis for the dynamics of the time-fractional KdV equation

In this paper, two classes of efficient difference schemes for the simulation of the time-fractional Korteweg–de Vries equation are carried out. The temporal derivative is approximated with the help of the uniform/nonuniform L1 formula and the uniform L2-1σ formula, respectively. The spatial derivative is done with the uniform centered difference scheme. Unique solvability, boundedness and convergence of the corresponding difference schemes are rigorously proved at length. As far as we know, the proposed schemes achieve the highest convergence order in space among all the difference methods for the time-fractional KdV equation. Several numerical examples confirm the theoretical results and demonstrate the dynamic behavior.

Unconditionally stable fully‐discrete finite element numerical scheme for active fluid model

AbstractIn this paper, we propose a linear, decoupled, unconditionally stable fully‐discrete finite element scheme for the active fluid model, which is derived from the gradient flow approach for an effective non‐equilibrium free energy. The developed scheme is employed by an implicit‐explicit treatment of the nonlinear terms and a second‐order Gauge–Uzawa method for the decoupling of computations for the velocity and pressure. We rigorously prove the unique solvability and unconditional stability of the proposed scheme. Several numerical tests are presented to verify the accuracy, stability, and efficiency of the proposed scheme. We also simulate the self‐organized motion under the various external body forces in 2D and 3D cases, including the motion direction of active fluid from disorder to order. Numerical results show that the scheme has a good performance in accurately capturing and handling the complex dynamics of active fluid motion.