Spectral computation of highly oscillatory integral equations in laser theory

Jing, Gao; Condon, Marissa; Iserles, Arieh

doi:10.1016/j.jcp.2019.06.045

Cited by 10 publications

(3 citation statements)

References 21 publications

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“…На часі є розробка математичних моделей з різними типами використання даних. Одним з прикладів інноваційних рішень стали нові інформаційні оператори, основні принципи побудови яких викладено в [1][2][3]. Чисельне інтегрування функцій багатьох змінних широко використовується у математичному моделюванні [4][5][6].…”

Section: наведено оптимальну за порядком точності кубатурну формулу н...unclassified

Optimal by the Order of Accuracy Cubature Formula for the Approximate Calculation of Triple Integrals from Fast Oscillating Functions in General View

Nechuiviter,

Ivanov

2023

Cybernetics and Computer Technologies

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Introduction. The rapid development of digital technologies encourages scientists to create new or improve existing mathematical models of technical processes. It is time to develop mathematical models with different types of data. In the tasks of digital signal and image processing, the approximate calculation of integrals from rapidly oscillating functions using new information operators makes it possible to build cubature formulas using different types of information: the values of functions on planes, lines and points can be used as data. The purpose is to present and investigate the optimal cubature formula for the approximate calculation of the triple integral from rapidly oscillating functions in the general form on the class of differential functions. Information about functions are traces on systems of mutually perpendicular planes. Results. The study of the problems of digital signal and image processing continued using the example of numerical integration of triple integrals from rapidly oscillating functions in the general form. The values of functions on systems of mutually perpendicular planes are using for constructed cubature formula. The main attention in the research focuses on obtaining the estimations of errors. Proposed cubature formula for the approximate calculation of the triple integral from rapidly oscillating functions in general view is optimal in order of accuracy on the class of differential functions. The conducted numerical experiment confirmed the theoretical results. Conclusions. The obtained results make it possible to build new and improve existing mathematical models of processes with different types of input information. New information operators are a powerful tool in the development of such models. Cubature formulas for the approximate calculation of integrals from rapidly oscillating functions of many variables have been created. Іn the construction of the formulas traces of the function on planes, lines, and points are used. Formulas in their construction use function traces on planes, lines, and points. In this work, a cubature formula for the approximate calculation of the triple integral from a rapidly oscillating function in the general form, which is optimal in order of accuracy, is constructed and investigated on the class of differentiable functions. A feature of the proposed formula is the use of values of functions on systems of mutually perpendicular planes as data. Keywords: integrals of rapidly oscillating functions of many variables, cubature formulas, new information operators, digital signal and image processing.

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Section: наведено оптимальну за порядком точності кубатурну формулу н...unclassified

Optimal by the Order of Accuracy Cubature Formula for the Approximate Calculation of Triple Integrals from Fast Oscillating Functions in General View

Nechuiviter,

Ivanov

2023

Cybernetics and Computer Technologies

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“…Many problems in mathematical physics (Hadjadj and Dussauge [19]), theory of elasticity (Abdou et al [6,7] and Popov [35]), contact problems in two layers of elastic materials (Bugami [9]), generalized potential theory (Alhazmi [11]), spectral relationships in laser theory (Gao et al [18]), quantum mechanics (Lienert and Tumulka [24]), and mixed problems in the idea of elasticity (Aleksandrovsk and Covalence [10] and Georgiadis and Gourgiotis [40]) lead to one kind of integral equation. As a result, we have discovered that integral equations (IEs) have tight ties to various subfields within many scientific disciplines.…”

Section: Introductionmentioning

confidence: 99%

A computational technique for computing second-type mixed integral equations with singular kernels

Mahdy,

Abdou,

Mohamed

2023

J. Math. Computer Sci.

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In the present article, we establish the numerical solution for the mixed Volterra-Fredholm integral equation (MV-FIE) and it has a discontinuous kernel in position. While the Volterra integral term is considered in the class of time C[0, T ], T < 1, and has a continuous kernel in time. The necessary conditions have been established to ensure that there is a single solution in the space L 2 [−1, 1] × C[0, T ], T < 1. By utilizing the separation of variables technique, MV-FIE is transformed to Fredholm integral equation (FIE) of the second kind with variables coefficients in time. The separation technique of variables helps the authors choose the appropriate time function to establish the conditions of convergence in solving the problem and obtaining its solution. Then, using the Boubaker polynomials method, we end up with a linear algebraic system (LAS) abbreviated. The Banach fixed point (BFP) hypothesis has been presented to determine the existence and uniqueness of the solution of the LAS. The convergence of the solution and the stability of the error are discussed. The Maple 18 software is used to perform some numerical calculations once some numerical experiments have been taken into consideration.

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“…The presence of integral equations in various sciences and their fields of applications led to different techniques of solutions. For example, in the contact problem (Abdou, et al, [1] , [2] ), fluid mechanics (Haggag and Dosqiyas [3] ), displacement problems in mechanics, and the theory of elasticity (Aleksandrovsk and Covalence [4] , Konstanda [5] , Georgiadis and Panos [6] ), and in laser theory (Gao, et al, [7] ). Due to the tremendous development in the various basic sciences, we find many analytical and numerical methods that have helped to solve many of these problems.…”

Section: Introductionmentioning

confidence: 99%

Solution of nonlinear mixed integral equation via collocation method basing on orthogonal polynomials

Jan¹

2022

Heliyon

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Spectral computation of highly oscillatory integral equations in laser theory

Cited by 10 publications

References 21 publications

Optimal by the Order of Accuracy Cubature Formula for the Approximate Calculation of Triple Integrals from Fast Oscillating Functions in General View

Optimal by the Order of Accuracy Cubature Formula for the Approximate Calculation of Triple Integrals from Fast Oscillating Functions in General View

A computational technique for computing second-type mixed integral equations with singular kernels

Solution of nonlinear mixed integral equation via collocation method basing on orthogonal polynomials

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