Problems For Differential Equations Research Articles

Вычисление коэффициентов определяющих соотношений нелинейно-упругих материалов с использованием эволюционных алгоритмов

Abstract. The possibility of parameter identification for constitutive relations of materials based on uniaxial stretching of a nonlinear-elastic prismatic sample and on inflating a hollow cylinder has been studied. The experiments are modeled using the semi-inverse method of nonlinear theory of elasticity. To describe the mechanical properties of the studied objects, the potentials of compressible nonlinearly elastic media - Blatz and Ko material, and Murnaghan model are used. In the first stage, a series of direct problems are solved. Analytical expressions for the load-elongation dependence are used for the stretching problem. The three-dimensional problem of a cylinder inflation is reduced to a nonlinear boundary value problem for an ordinary second-order differential equation for the function of the radial displacement of cylinder points. In all cases, load diagrams are constructed based on the specified parameters of the nonlinearly elastic models, which are subsequently considered as initial data for the inverse problem of parameter identification. The problem is reduced to finding the minimum of the objective function which is defined as a least squares function. Two evolutionary algorithms were used for analysis: the Squirrel search algorithm (SSA), based on the model of the foraging behaviour of southern flying squirrel population, and the differential evolution algorithm (DE) from the SciPy library. Reliable identification of material parameters of models was shown using evolutionary algorithms based on load diagrams of samples, including condition of artificial noise of "experimental" data.

THE TASK OF OPTIMIZING OIL LOADING IN TANKERS

The paper discusses the task of optimizing oil loading into tankers, using the Baku-Sufsi oil pipeline as an example. The goal is to develop an algorithm for building an optimization method for a tanker from an oil terminal, which represents the cost from the pipeline as a Fourier series. The problem of the hydraulic differential equation for loading oil flow from the terminal to the tankers is discussed. After integration, the equation does not have an obvious solution and must be solved approximately by an iterative or graphical method. As a result, factors and regularities are identified, which are combined with real oil parameters. Therefore, for the first time in Georgia, an original methodology for building an optimization algorithm for oil loading from a terminal to a tanker was developed.

Green’s function for solving initial-boundary value problem of evolutionary partial differential equations

We propose a new method to solve the initial-boundary value problem for hyperbolic-dissipative partial differential equations (PDEs) based on the spirit of LY algorithm [T.-P. Liu and S.-H. Yu, Dirichlet–Neumann kernel for hyperbolic-dissipative system in half-space, Bull. Inst. Math. Acad. Sin. 7 (2012) 477–543]. The new method can handle more general domains than that of LYs’. We convert the evolutionary PDEs into the elliptic PDEs by the Laplace transformation. Using the Laplace transformation of the fundamental solutions of the evolutionary PDEs and the image method, we can construct Green’s functions for the corresponding elliptic PDEs. Finally, we obtain Green’s functions for the evolutionary PDEs by inverting the Laplace transformation. As a consequence, we establish Green’s functions for some basic PDEs such as the heat equation, the wave equation and the damped wave equation, in a half space and a quarter plane with various boundary conditions. On the other hand, the structure of hyperbolic-dissipative PDEs means its fundamental solution is non-symestric and hence the image method does not work. We utilize the idea of Laplace wave train introduced by Liu and Yu in [Navier–Stokes equations in gas dynamics: Greens function, singularity and well-posedness, Comm. Pure Appl. Math. 75(2) (2022) 223–348] to generalize the image method. Combining this with the notions of Rayleigh surface wave operators introduced in [S. J. Deng, W. K. Wang and S.-H. Yu, Green’s functions of wave equations in [Formula: see text], Arch. Ration. Mech. Anal. 216 (2015) 881–903], we are able to obtain the complete representations of Green’s functions for the convection-diffusion equation and the drifted wave equation in a half space with various boundary conditions.

THE HEAT TRANSFER AND MAGNETOHYDRODYNAMICS PROBLEM WITH HEAT SOURCE IN HALF INFINITE 1-D DOMAIN

In this paper we consider the temperature and laminar flow of an incompressible conducting fluid past a non-conducting half-space. For the space approximation the finite differences method-finite difference scheme (FDS) and finite difference scheme with exact spectrum (FDSES) for solving the heat transfer and laminar flow initial boundary-value problem are used. This procedure allows reducing the problem to initial value problem for ordinary differential equations and the solution to the problem can be obtained numerically and analytically. The equation of the temperature is un-depending on the velocity and this function we can obtain in analytical form use the integral transform methods- Fourier and Laplace transforms.

Using Sadik Transform in Differential Equation

Differential equations are the equations used in engineering, science and physics. The differential equations which have linear differential equations and nonlinear differential equations. There are many ways to solve problems. Transformation is a way to find solutions to equations such as Laplace transform, Elzaki transform and Sumudu transform etc. In this research, Sadik transform is applied to solve the linear and nonlinear differential equation problems. In which nonlinear differential equation use homotopy perturbation to find solutions to equations and also present examples to illustrate the preciseness of this method.

Multiple Solutions for Lie´nard Type Generalized Equations

Two-point boundary value problems for second-order ordinary differential equations of Lie´nard type are studied. A comparison is made between equations x´´ + f (x) x´2 + g(x) = 0 and x´´ + f (x) x´ + g(x) = 0. In our approach, the Dirichlet boundary conditions are considered. The estimates of the number of solutions in both cases are obtained. These estimates are based on considering the equation of variations around the trivial solution and some additional assumptions. Examples and visualizations are supplied.

On one problem for a fourth-order mixed-type equation that degenerates inside and on the boundary of a domain

In the article, a nonlocal boundary value problem has been investigated for a fourth-order mixed-type equation degenerating inside and on the boundary of a domain. Applying the method of separation of variables to the problem under study, the spectral problem for an ordinary differential equation is obtained. The Green function of the last problem is constructed, with the help of which it is equivalently reduced to the Fredholm integral equation of the second kind with a symmetric kernel, which implies the existence of eigenvalues and the system of eigenfunctions for the spectral problem. The theorem of expansion of a given function into a uniformly convergent series with respect to the system of eigenfunctions is proved. Using the found integral equation and Mercer's theorem, a uniform convergence of some bilinear series depending on the found eigenfunctions is proved. The order of the Fourier coefficients is established. The solution of the problem under study is written as the sum of the Fourier series with respect to the system of eigenfunctions of the spectral problem. An estimate for the problem's solution is obtained, from which its continuous dependence on the given functions follows.

Gradient method with AFEM for parameter-estimation

We consider the adaptive finite element discretization of parameter estimation problems for nonlinear elliptic partial differential equations. The idea is to use a gradient method on the finite-dimensional parameter space for the minimization of the least-squares residual. Since the gradient involves solution of partial differential equations, it is not accesable, and is replaced by an approximation obtained by finite elements. This results into a perturbed gradient method. We use an (a posteriori) error estimator to control the accuracy of the gradient approximation and propose an algorithm, which links the estimator to the progress of the iteration. We show convergence of the algorithm under typical structural assumptions.

Algorithm for Approximate Solving of a Nonlinear Boundary Value Problem for Generalized Proportional Caputo Fractional Differential Equations

In this paper an algorithm for approximate solving of a boundary value problem for a nonlinear differential equation with a special type of fractional derivative is suggested. This derivative is called a generalized proportional Caputo fractional derivative. The new algorithm is based on the application of the monotone-iterative technique combined with the method of lower and upper solutions. In connection with this, initially, the linear fractional differential equation with a boundary condition is studied, and its explicit solution is obtained. An appropriate integral fractional operator for the nonlinear problem is constructed and it is used to define the mild solutions, upper mild solutions and lower mild solutions of the given problem. Based on this integral operator we suggest a scheme for obtaining two monotone sequences of successive approximations. Both sequences consist of lower mild solutions and lower upper solutions of the studied problem, respectively. The monotonic uniform convergence of both sequences to mild solutions is proved. The algorithm is computerized and applied to a particular example to illustrate the theoretical investigations.

Recursion Formulas for Integrated Products of Jacobi Polynomials

From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric series. With these contiguous relations one can prove several recursion formulas of those series. This theoretical result allows to compute integrals over products of Jacobi polynomials in a very efficient recursive way. Moreover, the authors present an application to numerical analysis where it can be used in algorithms which compute the approximate solution of boundary value problem of partial differential equations by means of the finite elements method. With the aid of the contiguous relations, the approximate solution can be computed much faster than using numerical integration. A numerical example illustrates this effect.

Integrated Stochastic Investigation of Singularly Perturbed Delay Differential Equations for the Neuronal Variability Model

The proposed research utilizes a computational approach to attain a numerical solution for the singularly perturbed delay differential equation (SPDDE) problem arising in the neuronal variability model through artificial neural networks (ANNs) with different solvers. The log-sigmoid function is used to construct the fitness function. The implementation of ANN on SPDDE problems is formulated for different solvers and trained with different weights. The optimization solvers such as the genetic algorithm (GA), sequential quadratic programming (SQP), and pattern search (PS) are hybridized with the active set technique (AST) and the interior-point technique (IPT) and is used to check the accuracy and rapid convergence of the numerical results of the SPDDE model. The numerical outcomes demonstrate that the system is easy to handle and efficient to solve with boundary conditions. Moreover, we used the mean residual error for one hundred runs for each solver to validate the accuracy of the proposed scheme.

Discrete Approximation of Solutions of the Cauchy Problem for a Linear Homogeneous Differential-Operator Equation with a Caputo Fractional Derivative in a Banach Space

In this paper, we construct and examine the time-discretization scheme for the Cauchy problem for a linear homogeneous differential equation with the Caputo fractional derivative of order α ∈ (0, 1) in time and containing the sectorial operator in a Banach space in the spatial part. The convergence of the scheme is established and error estimates are obtained in terms of the step of discretization. Properties of the Mittag-Leffler function, hypergeometric functions, and the calculus of sectorial operators in Banach spaces are used. Results of numerical experiments that confirm theoretical conclusions are presented.

Simultaneous digital twin identification and signal-noise decomposition through modified generalized sparse identification of nonlinear dynamics

A digital twin provides a digital replication of a physical system for remote monitoring, viewing, and control objectives. It has the potential to reshape the future of industrial processes, hence paving the way for smart manufacturing. Automatic system identification techniques that are robust to measurement noise are critical for the development of high-fidelity digital twins and their applications. By establishing a sparse regression framework, the sparse identification of nonlinear dynamics (SINDy) algorithm automatically determines the parsimonious governing equations for physical systems. However, there are some major challenges associated with using SINDy to identify digital twin models. First, the SINDy is restricted to solving the ordinary differential equation (ODE) and partial differential equation (PDE) problems. Second, measurement noise may significantly deteriorate the performance of SINDy. In this paper, the generalized SINDy (GSINDy) algorithm is first introduced to enlarge the SINDy’s applicable range. Then, the modified GSINDy (MGSINDy) algorithm is proposed, in which an objective function is constructed to simultaneously identify the digital twin input time-series dynamics model and output model while separating noise from the noisy input. Two numerical examples and one industrial case study are analysed to demonstrate the advantages of applying the proposed MGSINDy to construct digital twin models. Furthermore, the proposed algorithm can be integrated with the existing SINDy-based online model-adjusting frameworks to become online-adjustable.

Inverse problem for an abstract neutral differential equation of Sobolev-type

In this manuscript, we consider an inverse problem for the neutral differential equation of Sobolev type with a deviated argument and identify the parameter using an overdetermined condition on a mild solution. A direct approach using Volterra integral equations for sufficiently regular data and an optimal control approach for less regular data are the main techniques to find the result. Under certain hypotheses, the characterization of the limit of the sequence of approximate solutions demonstrates that it is a solution to the original inverse problem under specific hypotheses. At last, an example is provided in the support of our results.

Advances in Boundary Value Problems for Fractional Differential Equations

Fractional-order differential and integral operators and fractional differential equations have extensive applications in the mathematical modelling of real-world phenomena which occur in scientific and engineering disciplines such as physics, chemistry, biophysics, biology, medical sciences, financial economics, ecology, bioengineering, control theory, signal and image processing, aerodynamics, transport dynamics, thermodynamics, viscoelasticity, hydrology, statistical mechanics, electromagnetics, astrophysics, cosmology, and rheology [...]

On Solvability of Some Inverse Problems for a Fractional Parabolic Equation with a Nonlocal Biharmonic Operator

The paper considers the solvability of some inverse problems for fractional differential equations with a nonlocal biharmonic operator, which is introduced with the help of involutive transformations in two space variables. The considered problems are solved using the Fourier method. The properties of eigenfunctions and associated functions of the corresponding spectral problems are studied. Theorems on the existence and uniqueness of solutions to the studied problems are proved.

The approximate solution of nonlinear heat equation

AbstractNowadays there are many approximate methods for thermal conductivity calculation, that lead to satisfactory outcomes in engineering practice. With the new approach, the solution of the nonlinear thermal conductivity equation was investigated. In addition to this, we reviewed the approximate solution of the nonlinear equation of thermal conductivity with cubic nonlinearity. To solve the problems of mathematical analysis, differential and integral equations, and boundary value problems of mathematical physics, difference, and interpolation are applied. Thus, for thinking of the effectiveness and reasonableness of these approaches, it is crucial for their theoretical investigation. The solution to these questions was found for each class of equation and each of its methods in their way and was often represented as significant difficulty, and in many cases was an obstacle for the current time. A natural approach to solving this issue is the use of the ideas of functional analysis. The variational principle initially was considered as a variational approach for solving linear functional equations and finding eigenvalues of linear operators. As in any variational approach, the problem of solving an equation will be brought to finding the extremum of the certain function of a special type, given over the entire space. It was found that the approach is useful in a way of minimizing functions of the more general type.

Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type

We present an exponentially convergent numerical method to approximate the solution of the Cauchy problem for the inhomogeneous fractional differential equation with an unbounded operator coefficient and Caputo fractional derivative in time. The numerical method is based on the newly obtained solution formula that consolidates the mild solution representations of sub-parabolic, parabolic and sub-hyperbolic equations with sectorial operator coefficient A and non-zero initial data. The involved integral operators are approximated using the sinc-quadrature formulas that are tailored to the spectral parameters of A, fractional order α and the smoothness of the first initial condition, as well as to the properties of the equation’s right-hand side f(t). The resulting method possesses exponential convergence for positive sectorial A, any finite t, including t=0 and the whole range α∈(0,2). It is suitable for a practically important case, when no knowledge of f(t) is available outside the considered interval t∈[0,T]. The algorithm of the method is capable of multi-level parallelism. We provide numerical examples that confirm the theoretical error estimates.

Multigrid Incomplete Factorization Methods in Krylov Subspaces

The paper studies multigrid methods for solving systems of linear algebraic equations resulting from the seven-point discretization of the three-dimensional Dirichlet problem for an elliptic differential equation of the second order in a parallepipedal domain on a regular grid. The algorithms suggested are presented as special iteration processes of incomplete factorization in Krylov subspaces with a hierarchical recursive vector structure that corresponds to a sequence of embedded grids and gives rise to a block tridiagonal recursive representation of the coefficient matrix of the original linear algebraic system. The convergence of iterations is enhanced by using the principle of compensation of the row sums and also the symmetric successive block overrelaxation. An arbitrary m-grid method is defined recursively, based on the two-grid method. For simplicity, the algorithms are considered for linear systems with Stieltjes coefficient matrices. Issues related to generalization of the algorithms to larger classes of problems and, in particular, those with unsymmetric matrices are discussed.

Ergodic control of McKean–Vlasov SDEs and associated Bellman equation

We consider the ergodic control problem for McKean-Vlasov stochastic differential equations and prove the existence and uniqueness of the viscosity solution to the associated fully nonlinear HJB equation in a lifted sense. Furthermore, as the time horizon goes to infinity, we show that the solutions of finite-horizon time-averaging optimal control problems converge to that of the ergodic control problem. Our results require dissipativity conditions and dissipativity-like conditions on distribution variables of both drift and diffusion coefficients.