Problems For Differential Equations Research Articles

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.

Solvability of One Class of Inverse Problem for Partial Differential Equations

Solvability of One Class of Inverse Problem for Partial Differential Equations

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.

Boundary value problems for the harmonic differential equations of the third order

Boundary value problems for the harmonic differential equations of the third order

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.

The Eigenvalue Properties of a Kind of Singular Differential Equations in 3-Dimensional Space

In this paper, we consider the eigenvalue problem of the singular differential equation [see formula in PDF] in a bounded open ball with Dirichlet boundary condition in 3-dimensional space, where, [see formula in PDF][see formula in PDF], [see formula in PDF] is a given constant[see formula in PDF]. And we have made a detailed characterization of the weak solution space. Furthermore, the existence of the minimum eigenvalue and the fundamental gap are provided.

Three Boundary Value Problems for Complex Partial Differential Equations in the Lens Domain

Three Boundary Value Problems for Complex Partial Differential Equations in the Lens Domain

Solvability of the Generalized Ionkin Problem for Differential Equations with Multiple Characteristics

Solvability of the Generalized Ionkin Problem for Differential Equations with Multiple Characteristics

Numerical solutions of boundary problems in partial differential equations: A deep learning framework with Green's function

This work introduces Green Networks (GreenNets), a novel class of neural network architectures designed to address a wide spectrum of boundary problems in partial differential equations (PDEs). Demonstrated versatility includes solving the Poisson equation, convection-diffusion equation, and unsteady incompressible Navier-Stokes equations. GreenNets comprises two specialized variants: Boundary Transformation Green Network (BT-GreenNet), focusing on transforming Dirichlet boundary conditions into source terms, and Boundary Integral Green Network (BI-GreenNet), considering boundary integral terms with shared weights between the boundary and source integrals. Comparisons with existing deep learning and traditional numerical methods underscore their accuracy and efficiency. BT-GreenNet excels in efficiency, and BI-GreenNet offers enhanced flexibility. The research pinnacle, Boundary Unified Green Network (BU-GreenNet), merges BT-GreenNet and BI-GreenNet to address coupled PDEs, specifically unsteady incompressible Navier-Stokes equations. BU-GreenNet stands out by surpassing traditional numerical approaches, offering remarkable computational efficiency while faithfully capturing the intricate evolution of fluid dynamics phenomena. This research redefines the means of solution to boundary problems in PDEs through innovative neural network architectures, emphasizing practical utility and computational efficiency.

On Weighted Cauchy-Type Problem of Riemann-Liouville Fractional Differential Equations in Lebesgue Spaces with Variable Exponent

This paper aims to investigate the existence, uniqueness, and stability properties for a class of fractional weighted Cauchy-type problem in the variable exponent Lebesgue space $L^{p(.)}$. The obtained results are set up by employing generalized intervals and piece-wise constant functions so that the $L^{p(.)}$ is transformed into the classical Lebesgue spaces. Moreover, the usual Banach Contraction Principle is utilized, and the Ulam-Hyers (UH) stability is studied. At the final stage, we provide an example to support the accuracy of the obtained results.

Ulam Stability for Boundary Value Problems of Differential Equations—Main Misunderstandings and How to Avoid Them

Ulam type stability is an important property studied for different types of differential equations. When this type of stability is applied to boundary value problems, there are some misunderstandings in the literature. In connection with this, initially, we give a brief overview of the basic ideas of the application of Ulam type stability to initial value problems. We provide several examples with simulations to illustrate the main points in the application. Then, we focus on some misunderstandings in the application of Ulam stability to boundary value problems. We suggest a new way to avoid these misunderstandings and how to keep the main idea of Ulam type stability when it is applied to boundary value problems of differential equations. We present one possible way to connect both the solutions of the given problem and the solutions of the corresponding inequality. In addition, we provide several examples with simulations to illustrate the ideas for boundary value problems and we also show the necessity of the new way of applying the Ulam type stability. To illustrate the theoretical application of the suggested idea to Ulam type stability, we consider a linear boundary value problem for nonlinear impulsive fractional differential equations with the Caputo fractional derivative with respect to another function and piecewise-constant variable order. We define the Ulam–Hyers stability and obtain sufficient conditions on a finite interval. As partial cases, integral presentations of the solutions of boundary value problems for various types of fractional differential equations are obtained and their Ulam type stability is studied.