19 research outputs found
An Algorithm for the Closed-Form Solution of Certain Classes of Volterra–Fredholm Integral Equations of Convolution Type
No full textIn this paper, a direct operator method is presented for the exact closed-form solution of certain classes of linear and nonlinear integral Volterra–Fredholm equations of the second kind. The method is based on the existence of the inverse of the relevant linear Volterra operator. In the case of convolution kernels, the inverse is constructed using the Laplace transform method. For linear integral equations, results for the existence and uniqueness are given. The solution of nonlinear integral equations depends on the existence and type of solutions ofthe corresponding nonlinear algebraic system. A complete algorithm for symbolic computations in a computer algebra system is also provided. The method finds many applications in science and engineering
A Unified Formulation of Analytical and Numerical Methods for Solving Linear Fredholm Integral Equations
No full textThis article is concerned with the construction of approximate analytic solutions to linear Fredholm integral equations of the second kind with general continuous kernels. A unified treatment of some classes of analytical and numerical classical methods, such as the Direct Computational Method (DCM), the Degenerate Kernel Methods (DKM), the Quadrature Methods (QM) and the Projection Methods (PM), is proposed. The problem is formulated as an abstract equation in a Banach space and a solution formula is derived. Then, several approximating schemes are discussed. In all cases, the method yields an explicit, albeit approximate, solution. Several examples are solved to illustrate the performance of the technique
Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams
No full textRecent developments have shown that the widely used simplified differential model of Eringen’s nonlocal elasticity in nanobeam analysis is not equivalent to the corresponding and initially proposed integral models, the pure integral model and the two-phase integral model, in all cases of loading and boundary conditions. This has resolved a paradox with solutions that are not in line with the expected softening effect of the nonlocal theory that appears in all other cases. In addition, it revived interest in the integral model and the two-phase integral model, which were not used due to their complexity in solving the relevant integral and integro-differential equations, respectively. In this article, we use a direct operator method for solving boundary value problems for nth order linear Volterra–Fredholm integro-differential equations of convolution type to construct closed-form solutions to the two-phase integral model of Euler–Bernoulli nanobeams in bending under transverse distributed load and various types of boundary conditions
On the Exact Solution of Nonlocal Euler–Bernoulli Beam Equations via a Direct Approach for Volterra-Fredholm Integro-Differential Equations
No full textFirst, we develop a direct operator method for solving boundary value problems for a class of nth order linear Volterra–Fredholm integro-differential equations of convolution type. The proposed technique is based on the assumption that the Volterra integro-differential operator is bijective and its inverse is known in closed form. Existence and uniqueness criteria are established and the exact solution is derived. We then apply this method to construct the closed form solution of the fourth order equilibrium equations for the bending of Euler–Bernoulli beams in the context of Eringen’s nonlocal theory of elasticity (two phase integral model) under a transverse distributed load and simply supported boundary conditions. An easy to use algorithm for obtaining the exact solution in a symbolic algebra system is also given
A Comparative Study of Machine Learning Methods and Text Features for Text Authorship Recognition in the Example of Azerbaijani Language Texts
No full textThis paper presents various machine learning methods with different text features that are explored and evaluated to determine the authorship of the texts in the example of the Azerbaijani language. We consider techniques like artificial neural network, convolutional neural network, random forest, and support vector machine. These techniques are used with different text features like word length, sentence length, combined word length and sentence length, n-grams, and word frequencies. The models were trained and tested on the works of many famous Azerbaijani writers. The results of computer experiments obtained by utilizing a comparison of various techniques and text features were analyzed. The cases where the usage of text features allowed better results were determined
A Symbolic Method for Solving a Class of Convolution-Type Volterra–Fredholm–Hammerstein Integro-Differential Equations under Nonlocal Boundary Conditions
No full textIntegro-differential equations involving Volterra and Fredholm operators (VFIDEs) are used to model many phenomena in science and engineering. Nonlocal boundary conditions are more effective, and in some cases necessary, because they are more accurate measurements of the true state than classical (local) initial and boundary conditions. Closed-form solutions are always desirable, not only because they are more efficient, but also because they can be valuable benchmarks for validating approximate and numerical procedures. This paper presents a direct operator method for solving, in closed form, a class of Volterra–Fredholm–Hammerstein-type integro-differential equations under nonlocal boundary conditions when the inverse operator of the associated Volterra integro-differential operator exists and can be found explicitly. A technique for constructing inverse operators of convolution-type Volterra integro-differential operators (VIDEs) under multipoint and integral conditions is provided. The proposed methods are suitable for integration into any computer algebra system. Several linear and nonlinear examples are solved to demonstrate the effectiveness of the method
A Procedure for Factoring and Solving Nonlocal Boundary Value Problems for a Type of Linear Integro-Differential Equations
No full textThe aim of this article is to present a procedure for the factorization and exact solution of boundary value problems for a class of n-th order linear Fredholm integro-differential equations with multipoint and integral boundary conditions. We use the theory of the extensions of linear operators in Banach spaces and establish conditions for the decomposition of the integro-differential operator into two lower-order integro-differential operators. We also create solvability criteria and derive the unique solution in closed form. Two example problems for an ordinary and a partial intergro-differential equation respectively are solved
A Second-Order Numerical Method for a Class of Optimal Control Problems
No full textThe numerical solution of optimal control problems through second-order methods is examined in this paper. Controlled processes are described by a system of nonlinear ordinary differential equations. There are two specific characteristics of the class of control actions used. The first one is that controls are searched for in a given class of functions, which depend on unknown parameters to be found by minimizing an objective functional. The parameter values, in general, may be different at different time intervals. The second feature of the considered problem is that the boundaries of time intervals are also optimized with fixed values of the parameters of the control actions in each of the intervals. The special cases of the problem under study are relay control problems with optimized switching moments. In this work, formulas for the gradient and the Hessian matrix of the objective functional with respect to the optimized parameters are obtained. For this, the technique of fast differentiation is used. A comparison of numerical experiment results obtained with the use of first- and second-order optimization methods is presented
Closed-Form Solutions of Linear Ordinary Differential Equations with General Boundary Conditions
No full textThis paper deals with the solution of boundary value problems for ordinary differential equations with general boundary conditions. We obtain closed-form solutions in a symbolic form of problems with the general n-th order differential operator, as well as the composition of linear operators. The method is based on the theory of the extensions of linear operators in Banach spaces
Evaluating Landscape Fragmentation and Consequent Environmental Impact of Solar Parks Installation in Natura 2000 Protected Areas: The Case of the Thessaly Region, Central Greece
No full textThis study examines the adverse environmental impacts of solar photovoltaic parks located in established protected areas, aiming to determine the level of landscape fragmentation through the calculation of relevant landscape metrics. For this purpose, a case study was carried out in a Mediterranean Natura 2000 Special Protection Area (SPA), and landscape metrics were calculated using Geographic Information System spatial analysis tools. The analysis of metrics showed that the installation of renewable energy parks within the designated protected area negatively affect landscape fragmentation and the absence of carefully defined and evidence-based mitigation measures. The land cover categories that are significantly affected are those considered critical habitats of bird species that have been designated as SPAs. The results of this study highlight the need to integrate, in the National Renewable Energy Spatial Plans, specific biodiversity objectives, such as conservation objectives and the suspension of the installation of photovoltaic parks in certain areas that are important for conservation of biodiversity, in order to ensure the overall sustainability of renewable energy production
