Problems For Differential Equations Research Articles

Distribution of eigenvalues of a perturbed differentiation operator on the interval

In this paper, we construct a characteristic determinant of the spectral problem of a first-order differential equation on an interval with an integral perturbation in the boundary value condition, which is an entire analytic function of the spectral parameter. Based on the formula for the characteristic determinant, conclusions are drawn about the asymptotic behavior of the spectrum of the perturbed spectral problem depending on the modulus of continuity of the subinteral function.

Semilinear optimal control with Dirac measures

Abstract The purpose of this work is to study an optimal control problem for a semilinear elliptic partial differential equation with a linear combination of Dirac measures as a forcing term; the control variable corresponds to the amplitude of such singular sources. We analyze the existence of optimal solutions and derive first- and, necessary and sufficient, second-order optimality conditions. We develop a solution technique that discretizes the state and adjoint equations with continuous piecewise linear finite elements; the control variable is already discrete. We analyze the convergence properties of discretizations and obtain, in two dimensions, an a priori error estimate for the underlying approximation of an optimal control variable.

On the solvability of boundary value problems for iterative fractional differential equations

On the solvability of boundary value problems for iterative fractional differential equations

Boundary value problems of quaternion-valued differential equations: solvability and Green’s function

This paper is associated with Sturm–Liouville type boundary value problems and periodic boundary value problems for quaternion-valued differential equations (QDEs). Employing the theory of quaternionic matrices, we prove the conditions for the solvability of the linear boundary valued problem and find Green’s function.

Existence and stability results for the integrable solution of a singular stochastic fractional-order integral equation with delay

In this paper, we are concerning with the existence of the solution \( \V \in L_1([0,\tau],L_2(\Omega))\) of the singular stochastic fractional-order integral equation with delay \(\varrho(.) \), \[ \V(t) = B(t) t^{\alpha - 1} + \lambda ~ \I^{\beta} \G(t,\V(\varrho (t))), ~~~t\in (0,\tau], \] where \(B(t)\) is a given second order mean square stochastic process, \( \lambda \) is a parameter, \(\varrho (t) \leq t\), and \(\G(t,\V) \) is a measurable function in \(t \in (0,\tau]\) and satisfies Lipschitz condition on the second argument. %and \(x\in C_{1-\beta}([0,T],L_2(\Omega)) $ will be proved The Hyers-Ulam and generalized Hyers-Ulam-Rassias stability will be proved. Moreover, the continuous dependence of the solution on the process \(B(t)\) and \(\lambda\) will be studied. As applications, some nonlocal, weighted and nonlocal-weighted integral problems of stochastic fractional-order differential equations will be studied.

On the numerical integration of singular initial and boundary value problems for generalised Lane–Emden and Thomas–Fermi equations

We propose a geometric approach for the numerical integration of singular initial and boundary value problems for (systems of) quasi-linear differential equations. It transforms the original problem into the problem of computing the unstable manifold at a stationary point of an associated vector field and thus into one which can be solved in an efficient and robust manner. Using the shooting method, our approach also works well for boundary value problems. As examples, we treat some (generalised) Lane–Emden equations and the Thomas–Fermi equation.

Non-centered parametric variational Bayes’ approach for hierarchical inverse problems of partial differential equations

This paper proposes a non-centered parameterization based infinite-dimensional mean-field variational inference (NCP-iMFVI) approach for solving the hierarchical Bayesian inverse problems. This method can generate available estimates from the approximated posterior distribution efficiently. To avoid the mutually singular obstacle that occurred in the infinite-dimensional hierarchical approach, we propose a rigorous theory of the non-centered variational Bayesian approach. Since the non-centered parameterization weakens the connection between the parameter and the hyper-parameter, we can introduce the hyper-parameter to all terms of the eigendecomposition of the prior covariance operator. We also show the relationships between the NCP-iMFVI and infinite-dimensional hierarchical approaches with centered parameterization. The proposed algorithm is applied to three inverse problems governed by the simple smooth equation, the Helmholtz equation, and the steady-state Darcy flow equation. Numerical results confirm our theoretical findings, illustrate the efficiency of solving the iMFVI problem formulated by large-scale linear and non-linear statistical inverse problems, and verify the mesh-independent property.

Biorthogonal flatlet multiwavelet collocation method for solving the singular nonlinear system with initial and boundary conditions

PurposeThe numerical methods are of great importance for approximating the solutions of a system of nonlinear singular ordinary differential equations. In this paper, the authors present the biorthogonal flatlet multiwavelet collocation method (BFMCM) as a numerical scheme for a class of system of Lane–Emden equations with initial or boundary or four-point boundary conditions.Design/methodology/approachThe approach is involved in combining the biorthogonal flatlet multiwavelet (BFM) with the collocation method. The authors investigate the properties and procedure of the BFMCM for first time on this class of equations. By using the BFM and the collocation points, the method is constructed and it transforms the nonlinear differential equations problem into a system of nonlinear algebraic equations. The unknown coefficients of the assuming solution are determined by solving the obtained system. Additionally, convergence analysis and numerical stability of the suggested method are provided.FindingsAccording to the attained results, the proposed BFMCM has more accurate results in comparison with results of other methods. The maximum absolute errors are calculated by using the BFMCM for comparison purposes provided.Originality/valueThe key desirable properties of BFMCM are its efficiency, simple applicability and minimizes errors. Therefore, the proposed method can be used to solve nonlinear problems or problems with singular points.

Application of neural network modeling methods in solving initial boundary value problems for partial differential equations

Machine learning allows you to solve a variety of data analysis problems, but its use for solving differential equations has appeared relatively recently. The approximation of the solution of the boundary value problem for differential equations (ordinary and partial derivatives) is constructed using neural network functions. The selection of weighting coefficients is carried out during the training of the neural network. The criteria for the quality of training in this case are inconsistencies in the equation and boundary-initial conditions. This approach makes it possible, instead of grid solutions, to find solutions defined on the entire feasible region of the boundary value problem. Specific examples show the features of the application of physics-informed neural networks to the solution of boundary value problems for differential equations of various types. Physics-informed neural networks training methods can be used in the tasks of retraining intelligent control systems on incomplete sets of input data.

Results on Ulam‐type stability of linear differential equation with integral transform

The main theme of this study is to implement the Sumudu integral transform technique to solve the stability problem of linear differential equations. Another important aspect of this paper is to investigate the Ulam–Hyers and Ulam–Hyers–JRassias stability of linear differential equations by using Sumudu transform method. Further, the results are extended to the Mittag‐Leffler–Ulam–Hyers and Mittag‐Leffler–Ulam–Hyers–JRassias stability of these differential equations. As an application point of view, the Sumudu transform is used to find Ulam stabilities of differential equations arising in field‐controlled DC servo motor with position control.

PyPESTO: a modular and scalable tool for parameter estimation for dynamic models.

Mechanistic models are important tools to describe and understand biological processes. However, they typically rely on unknown parameters, the estimation of which can be challenging for large and complex systems. pyPESTO is a modular framework for systematic parameter estimation, with scalable algorithms for optimization and uncertainty quantification. While tailored to ordinary differential equation problems, pyPESTO is broadly applicable to black-box parameter estimation problems. Besides own implementations, it provides a unified interface to various popular simulation and inference methods. pyPESTO is implemented in Python, open-source under a 3-Clause BSD license. Code and documentation are available on GitHub (https://github.com/icb-dcm/pypesto).

On the solvability of a nonlocal problem for a Boussinesq-type differential equation

We study a nonlocal problem for a differential Boussinesq-type equations in a multidimensional domain. Conditions for the existence and uniqueness of the solution are established, and a spectral decomposition of the solution is obtained.

On Local Unique Solvability for a Class of Nonlinear Identification Problems

Nonlinear identification problems for evolution differential equations, solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are studied. An equation of the considered class contains a linear unbounded operator, which generates analytic resolving families for the corresponding linear homogeneous equation, and a continuous nonlinear operator, which depends on lower-order Dzhrbashyan–Nersesyan derivatives and a depending on time unknown element. The identification problem consists of the equation, Dzhrbashyan–Nersesyan initial value conditions and an abstract overdetermination condition, which is defined by a linear continuous operator. Using the contraction mappings theorem, we prove the unique local solvability of the identification problem. The cases of mild and classical solutions are studied. The obtained abstract results are applied to an investigation of a nonlinear identification problem to a linearized phase field system with time dependent unknown coefficients at Dzhrbashyan–Nersesyan time-derivatives of lower orders.

Random differential hyperbolic equations of fractional order in Fréchet spaces

Abstract In the present paper, we provide some existence results for the Darboux problem of partial fractional random differential equations in Fréchet spaces with an application of a generalization of the classical Darbo fixed point theorem and the concept of measure of noncompactness.

An Operator Method for Investigation of the Stability of Time-Dependent Source Identification Telegraph Type Differential Problems

This article is devoted to the study of the stability of time-dependent source identification telegraph type differential problems with dependent coefficients. Time-dependent source identification problems (SIPs) for telegraph differential equations (TDEs) with constant coefficients can be solved by classical integral-transform methods. However, these classical methods can be used, basically, in cases where the differential equation has constant coefficients. We establish the basic theorem of the stability of the time-dependent SIPs for the second-order linear differential equation (DE) in a Hilbert space with a self-adjoint positive definite operator (SAPDO) and damping term. In practice, stability estimates for the solution of the three types of SIPs for one-dimensional and for multidimensional TDEs with dependent coefficients and classic and non-classic conditions are obtained.

Spectral Analysis of One Class of Positive-Definite Operators and Their Application in Fractional Calculus

This paper is devoted to the spectral analysis of one class of integral operators, associated with the boundary-value problems for differential equations of fractional order. In particular, we show the positive definiteness of studying operators, which makes it possible to select areas in the complex plane where there are no eigenvalues for these operators.

Analytical solution of the space-time fractional reaction-diffusion equation with variable coefficients

In this paper, we solve the problem of an inhomogeneous one-dimensional fractional differential reaction-diffusion equation with variable coefficients (1.1)-(1.2) by the method of separation of variables (the Fourier method). The Caputo derivative and the Riemann-Liouville derivative are considered in the time and space directions, respectively. We prove that the obtained solution of the boundary-value problem satisfies the given boundary conditions. We discuss the convergence of the series defining the proposed solution.

Differential Evolution Using Enhanced Mutation Strategy Based on Random Neighbor Selection

Symmetry in a differential evolution (DE) transforms a solution without impacting the family of solutions. For symmetrical problems in differential equations, DE is a strong evolutionary algorithm that provides a powerful solution to resolve global optimization problems. DE/best/1 and DE/rand/1 are the two most commonly used mutation strategies in DE. The former provides better exploitation while the latter ensures better exploration. DE/Neighbor/1 is an improved form of DE/rand/1 to maintain a balance between exploration and exploitation which was used with a random neighbor-based differential evolution (RNDE) algorithm. However, this mutation strategy slows down convergence. It should achieve a global minimum by using 1000 × D, where D is the dimension, but due to exploration and exploitation balancing trade-offs, it can not achieve a global minimum within the range of 1000 × D in some of the objective functions. To overcome this issue, a new and enhanced mutation strategy and algorithm have been introduced in this paper, called DE/Neighbor/2, as well as an improved random neighbor-based differential evolution algorithm. The new DE/Neighbor/2 mutation strategy also uses neighbor information such as DE/Neighbor/1; however, in addition, we add weighted differences after various tests. The DE/Neighbor/2 and IRNDE algorithm has also been tested on the same 27 commonly used benchmark functions on which the DE/Neighbor/1 mutation strategy and RNDE were tested. Experimental results demonstrate that the DE/Neighbor/2 mutation strategy and IRNDE algorithm show overall better and faster convergence than the DE/Neighbor/1 mutation strategy and RNDE algorithm. The parametric significance test shows that there is a significance difference in the performance of RNDE and IRNDE algorithms at the 0.05 level of significance.

An investigation of an inverse problem for second-order abstract differential equation

An investigation of an inverse problem for second-order abstract differential equation

Component Mode Synthesis Based on the Energy Method for Frequency Domain Dynamic Response Analysis of Ballastless Track Structure

Used extensively in structural dynamics analysis, the energy method is advantageous for transforming boundary value problems of differential equations into variational extremum problems. Recent applications include analyzing the wave and vibration characteristics of track structures. However, traditional energy methods for dynamic modeling and analysis of ballastless track structures require obtaining the global stiffness matrix and mass matrix of the entire coupled system, which decreases computational efficiency. The EM-CMS algorithm for frequency domain dynamic response analysis of ballastless track structures has been proposed to address this issue. The core of EM-CMS is the development of model reduction strategies within the framework of the energy method to reduce matrix dimensions and thereby improve computational efficiency. Specifically, the steel spring floating slab track is the focus of this research. Utilizing the energy functional variational method, the modal properties of the rail and floating slab structures are obtained, and truncation is performed. A reduced-order model for the steel spring floating slab track is established by considering the boundary conditions between the rail-floating slab and the floating slab foundation (connected through fastener springs and steel springs, respectively, and considering the springs’ elastic potential energy). Comparatively, the computational efficiency of EM-CMS is approximately ten times greater than that of the energy method. The method’s accuracy is also carefully validated against finite element simulations.