Problems For Differential Equations Research Articles

Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain

In this paper, the direct segregated Boundary Domain Integral Equations (BDIEs) for the Mixed Boundary Value Problems (MBVPs) for a scalar second order elliptic Partial Differential Equation (PDE) with variable coefficient in unbounded (exterior) 2D domain is considered. Otar Chkadua, Sergey Mikhailov and David Natroshvili formulated both the interior and exterior 3D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order elliptic PDE with a variable coefficients. On the other hand Sergey Mikhailov and Tamirat Temesgen formulated only the interior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. However, in this paper we formulated the exterior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. The aim of this work is to reduce the MBVPs to some direct segregated BDIEs with the use of an appropriate parametrix (Levi function). We examine the characteristics of corresponding parametrix-based integral volume and layer potentials in some weighted Sobolev spaces, as well as the unique solvability of BDIEs and their equivalence to the original MBVPs. This analysis is based on the corresponding properties of the MBVPs in weighted Sobolev spaces that are proved as well.

Boundary problems of sequential fractional differential equations having a monomial coefficient

The article examines sequential fractional-order differential equations composed of monomial coefficients defined as nonlinear boundary problems. The original problem is converted into an integral equation in an equivalent form. The contraction mapping principle and Krasnoselskii's fixed-point theorem are implemented to obtain two existence conditions. The problem of Ulam-Hyers stability is also investigated. An illustration is presented to showcase practical application of the obtained results.

Optimal controls for forward-backward stochastic differential equations: Time-inconsistency and time-consistent solutions

This paper is concerned with an optimal control problem for a forward-backward stochastic differential equation (FBSDE, for short) with a recursive cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). It is found that such an optimal control problem is time-inconsistent in general, even if the cost functional is reduced to a classical Bolza type one as in Peng [47], Lim–Zhou [38], and Yong [72]. Therefore, instead of finding a global optimal control (which is time-inconsistent), we will look for a time-consistent and locally optimal equilibrium strategy, which can be constructed via the solution of an associated equilibrium Hamilton–Jacobi–Bellman (HJB, for short) equation. A verification theorem for the local optimality of the equilibrium strategy is proved by means of the generalized Feynman–Kac formula for BSVIEs and some stability estimates of the representation parabolic partial differential equations (PDEs, for short). Under certain conditions, it is proved that the equilibrium HJB equation, which is a nonlocal PDE, admits a unique classical solution. As special cases and applications, the linear-quadratic problems, a mean-variance model, a social planner problem with heterogeneous Epstein–Zin utilities, and a Stackelberg game are briefly investigated. It turns out that our framework can cover not only the optimal control problems for FBSDEs studied in [47,38,72], and so on, but also the problems of the general discounting and some nonlinear appearance of conditional expectations for the terminal state, studied in Yong [73,75] and Björk–Khapko–Murgoci [6].

Boundary value problems for integro-differential and singular higher-order differential equations

Abstract We investigate third-order strongly nonlinear differential equations of the type ( Φ ( k ( t ) u ″ ( t ) ) ) ′ = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t){u}^{^{\prime\prime} }\left(t)))^{\prime} =f\left(t,u\left(t),u^{\prime} \left(t),{u}^{^{\prime\prime} }\left(t)),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], where Φ \Phi is a strictly increasing homeomorphism, and the non-negative function k k may vanish on a set of measure zero. Using the upper and lower solution method, we prove existence results for some boundary value problems associated with the aforementioned equation. Moreover, we also consider second-order integro-differential equations like ( Φ ( k ( t ) v ′ ( t ) ) ) ′ = f t , ∫ 0 t v ( s ) d s , v ( t ) , v ′ ( t ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t)v^{\prime} \left(t)))^{\prime} =f\left(t,\underset{0}{\overset{t}{\int }}v\left(s){\rm{d}}s,v\left(t),v^{\prime} \left(t)\right),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], for which we provide existence results for various types of boundary conditions, including periodic, Sturm-Liouville, and Neumann-type conditions.

Numerical treatment of singular ODEs using finite difference and collocation methods

Boundary value problems (BVPs) in ordinary differential equations (ODEs) with singularities arise in numerous mathematical models describing real-life phenomena in natural sciences and engineering. This motivates vivid research activities aiming to characterize the analytical properties of singular problems, to investigate convergence of the standard numerical methods when they are applied to simulate differential equation with singularities, and to provide software for their efficient numerical treatment. There are two well-known, high order numerical methods which we focus on in this paper, the finite difference schemes and the collocation methods. Those methods proved to be dependable and highly accurate in the context of regular differential equations, so the question arises how do they preform for singular problems. While, there is a strong evidence for the collocation schemes to be a robust method to solve singular systems in a stable and efficient way, finite difference schemes are still considered less suitable for this problem class.In this paper, we shall compare the performance of the code HOFiD_bvp based on the high order finite difference schemes and bvpsuite2.0 based on polynomial collocation, when the codes are applied to singular problems in ODEs. We are fully aware of the difficulties in a code comparison, so in this paper, we will try to only diagnose the potential improvements, we could address in the next update of the codes.

Two-Point Boundary-Value Problem for Linear Stochastic Differential Equations

Two-Point Boundary-Value Problem for Linear Stochastic Differential Equations

Nonparametric estimation for periodic stochastic differential equations driven by [formula omitted]-Brownian motion

The paper is concerned with the nonparametric estimation problem for periodic stochastic differential equations driven by G-Brownian motion based on continuous observations. The consistency and asymptotic distribution of the nonparametric estimator are discussed. Computer simulations are performed to illustrate our theory.

A Birkhoff–Kellogg Type Theorem for Discontinuous Operators with Applications

By means of fixed point index theory for multivalued maps, we provide an analogue of the classical Birkhoff–Kellogg Theorem in the context of discontinuous operators acting on affine wedges in Banach spaces. Our theory is fairly general and can be applied, for example, to eigenvalues and parameter problems for ordinary differential equations with discontinuities. We illustrate in detail this fact for a class of second-order boundary value problem with deviated arguments and discontinuous terms. In a specific example, we explicitly compute the terms that occur in our theory.

Existence of Eigenvalues and Boundedness of the Eigenfunctions and the Orthogonality of the Eigenfunctions of a Type of Spectral Problem

Background: This research aims to investigate certain requirements for the presence of eigenvalues, as well as the boundaries of eigenfunctions and their derivatives, specifically, the eigenfunctions' first as well as second derivatives. Materials and Methods: In this study, we use the spectral problem of second-order differential equations: , with mixed boundary conditions , where is a spectral parameter. and the normalized condition , where . Results: We get that the spectral parameter of a second-order differential operator is real. And we obtain Lagrange’s identity for a spectral problem. Also, we prove that the spectral problem is self-adjoint, and the property of orthogonality of eigenfunctions is shown. Conclusions: In this research, we studied the existence of eigenvalues and the estimation of the norm of eigenfunctions for problem (1) - (3). Furthermore, we investigated the self-adjoint nature of the problem, and we proved that the eigenfunctions are orthogonal.

Numerical study on nanofluidic transport mechanism in Ellis flow within curved channel comprising compliant walls subjected to peristaltic activity

Peristaltic movement of fluid flows has significant applications in biomedical engineering, medicine, human physiology, etc. Specifically, it is very useful to understand and cure the very common intestinal diseases in human beings. A number of theoretical and empirical models are used to analyze peristaltic movement. In this work, the peristaltic movement of nanofluid is modeled with a non-Newtonian Ellis fluid model in a curved channel with compliant wall properties. The effects of Brownian motion, thermophoresis, and nonlinear radiations are considered in the heat transfer for better thermal analysis. The mathematical modeling of the physical problem yields the nonlinear partial differential equations with boundary conditions. First, the governing partial differential equations are non-dimensionalized, and then the resultant system is simplified by using the assumptions of a small Reynolds number and long wavelength. Then the obtained boundary value problem of differential equations is solved with the built-in Mathematica command NDSolve. The accuracy and reliability of the adopted procedure are verified by comparing the computed results with the reported literature. The impacts of the pertinent parameters (Brownian motion, thermal radiation, mixed convection, and thermophoresis phenomenon) on thermal energy, velocity, concentration, heat transfer rate, and stress at the lower wall are analyzed both in qualitative and quantitative manners. This study revealed some interesting facts, such as the peristaltic-driven motion of nanoliquid is strongly influenced by wall properties (i.e., wall elasticity, mass density, and wall damping). In addition, the flow experienced more resistance in the case of larger wall damping, but larger wall elasticity and mass density provide favorable movement for fluid motion. In addition, mixed convection plays a vital role in heat transfer and nanoparticle concentration in the curved domain. In addition, the curved channel walls have a higher stress factor than straight-plane channels. The results of the current study are very useful to understand many biological phenomena, such as the peristaltic movement of liquid during dialysis, food movement through the intestine, etc.

An efficient numerical method for modeling silver powder heat exchanger in dilution refrigerator

Dilution refrigerator provides continuous ultralow temperature environments as low as 10 mK. It has been widely used in a variety of important applications such as quantum computations. The silver powder heat exchanger in a dilution refrigerator plays a crucial role in realizing ultralow temperatures on the order of 10 mK by precooling the circulation of helium mixtures. To study the silver powder heat exchanger quantitatively, we have proposed an efficient numerical solution method for its thermodynamic model. This method utilizes constraint conditions cleverly to convert the initial value problem of differential equations into a boundary value problem, allowing us to solve the control equations using the existing ODE function quickly. Furthermore, we demonstrate the application of this method in the evaluation and design of silver powder heat exchanger. The research results of this paper have certain significance for the development of dilution refrigerator.

Diagonally quadratic BSDE with oblique reflection and optimal switching

The present paper is devoted to the study of diagonally quadratic backward stochastic differential equation with oblique reflection. Using a penalization approach, we show the existence of a solution by providing some delicate a priori estimates. We further obtain the uniqueness by verifying the first component of the solution is indeed the value of a switching problem for quadratic BSDEs. Moreover, we provide an extension for the solvability and apply our results to study a risk-sensitive switching problem for functional stochastic differential equations.

Solutions of periodic boundary value problems for first-order linear fuzzy differential equations under new conditions

Solutions of periodic boundary value problems for first-order linear fuzzy differential equations under new conditions

Inverse Coefficient Problem for the 2D Wave Equation with Initial and Nonlocal Boundary Conditions

In this paper, we consider direct and inverse problems for the two-dimensional wave equation. The direct problem is an initial boundary value problem for this equation with nonlocal boundary conditions. In the inverse problem, it is required to find the time-variable coefficient at the lower term of the equation. The classical solution of the direct problem is presented in the form of a biorthogonal series in eigenvalues and associated functions, and the uniqueness and stability of this solution are proven. For solution to the inverse problem, theorems of existence in local, uniqueness in global, and an estimate of conditional stability are obtained. The problems of determining the right-hand sides and variable coefficients at the lower terms from initial boundary value problems for second-order linear partial differential equations with local boundary conditions have been studied by many authors. Since the nonlinearity is convolutional, the unique solvability theorems in them are proven in a global sense. In the works, the method of separation of variables is used to find the classical solution of the direct problem in the form of a biorthogonal series in terms of eigenfunctions and associated functions. The nonlocal integral condition is used as the overdetermination condition with respect to the solution of the direct problem. The direct problem reduces to equivalent integral equations of the Fourier method. To establish integral inequalities, the generalized Gronwall-Bellman inequality is used. We obtain an a priori estimate of the solution in terms of an unknown coefficient, that are useful for studying the inverse problem.

Properties of eigenfunctions of a boundary value problem for ordinary differential equations of fourth-order with boundary conditions depending on the spectral parameter

In this paper, we consider an eigenvalue problem for fourth-order ordinary differential equations with a spectral parameter contained in all boundary conditions. We characterize the location of eigenvalues on the real axis, find their multiplicities, and study the oscillatory properties of eigenfunctions. Moreover, we obtain refined asymptotic formulas for the eigenvalues, as well as for the values at the endpoints of the interval of eigenfunctions and their derivatives. Then, using these results and the operator-theoretic formulation, we establish sufficient conditions for the system of eigenfunctions of this problem to form a basis in the space Lp,1<p<∞, after removing four functions.

Stochastic maximum principle for partially observed optimal control problem of McKean–Vlasov FBSDEs with Teugels martingales

Abstract In this paper, we study the stochastic maximum principle for a partially observed optimal control problem of forward-backward stochastic differential equations (FBSDEs for short) of McKean–Vlasov type driven by both a family of Teugels martingales and an independent Brownian motion. The coefficients of the system and the cost functional depending on the state of the solution process as well as its probability law and the control variable. We establish partially observed necessary conditions of optimality for this system under assumption that the control domain is supposed to be convex. Our main result is based on Girsavov’s theorem and the derivatives with respect to probability law. As an application of the general theory, a partially observed linear-quadratic control problem of McKean–Vlasov type is studied in terms of stochastic filtering.

Triple Mixed Integral Transformation and Applications for Initial-Boundary Value Problems

In this paper, we establish a new integral transformation method for solving the initial-boundary value problems of the partial differential equations. The method is a combined expression mainly based on the triple integral transformation of Laplace and Sumudu. We study the basic properties of the triple Laplace-Sumudu transform, and give the triple Laplace-Sumudu transforms of some basic functions. And as applications, we solve some heat flow equations and wave equations with initial-boundary value conditions by using the method. The results show that the triple Laplace-Sumudu transform is more efficient and useful to deal with these problems.

Two-point boundary-value problems for differential equations with generalized piecewise-constant argument

UDC 517.9 We consider a two-point boundary-value problem for a system of differential equations with generalized piecewise-constant argument. To solve the problem, we propose to use a constructive method based on the Dzhumabaev parametrization method and a new approach to the concept of general solution. The interval is partitioned with regard for the singularities of the argument. The values of the solution at the interior points of the partition are regarded as additional parameters, and the differential equation is transformed into a system of Cauchy problems with parameters on subintervals of the partition. By using the solutions of these problems, we obtain a new general solution of the differential equation with piecewise-constant argument and establish its properties. The new general solution, boundary conditions, and the conditions of continuity of the solution at the interior points of the partition are used to construct a linear system of algebraic equations for the introduced parameters. The coefficients and the right-hand side of the system are found as a result of the solution of Cauchy problems for linear ordinary differential equations on the subintervals of the partition. It is shown that the solvability of the boundary-value problem is equivalent to the solvability of the constructed system. We propose algorithms of the parametrization method for solving the analyzed boundary-value problem and establish necessary and sufficient conditions for the well-posedness of this problem.

Direct and inverse problems for the Hilfer fractional differential equation

The article studies direct and inverse problems for subdiffusion equations involving a Hilfer fractional derivative. An arbitrary positive self-adjoint operator $A$ is taken as the elliptic part of the equation. In particular, as the operator $A$ we can take the Laplace operator with the Dirichlet condition. First, the existence and uniqueness of a solution to the direct problem is proven. Then, using the representation of the solution to the direct problem, the existence and uniqueness of the inverse problem of finding the right-hand side of the equation, which depends only on the spatial variable, is proved.

ON THE FORMULATION OF BOUNDARY VALUE PROBLEMS FOR ONE SECOND-ORDER EQUATION

This paper addresses the formulation of boundary value problems (BVPs) for second-order differential equations. Boundary value problems are essential in various scientific and engineering applications where solutions must satisfy specific conditions at the boundaries of the domain. The study outlines a systematic approach to defining the differential equation, determining the domain, and specifying the appropriate boundary conditions. The discussion includes different types of boundary conditions such as Dirichlet, Neumann, and mixed conditions. An example is provided to illustrate the formulation process, demonstrating how to combine the differential equation with boundary conditions to define a complete BVP. Methods for solving these problems, including analytical and numerical techniques, are also reviewed, highlighting their importance in obtaining accurate solutions for complex systems.