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Journal articles on the topic 'Integral and its inverse operator'

Author: Grafiati

Published: 4 June 2025

Last updated: 1 August 2025

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1

Ali, Rana Safdar, Saba Batool, Shahid Mubeen, et al. "On generalized fractional integral operator associated with generalized Bessel-Maitland function." AIMS Mathematics 7, no. 2 (2022): 3027–46. http://dx.doi.org/10.3934/math.2022167.

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<abstract><p>In this paper, we describe generalized fractional integral operator and its inverse with generalized Bessel-Maitland function (BMF-Ⅴ) as its kernel. We discuss its convergence, boundedness, its relation with other well known fractional operators (Saigo fractional integral operator, Riemann-Liouville fractional operator), and establish its integral transform. Moreover, we have given the relationship of BMF-Ⅴ with Mittag-Leffler functions.</p></abstract>

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2

Ali, R. S., S. Mubeen, I. Nayab, Serkan Araci, G. Rahman, and K. S. Nisar. "Some Fractional Operators with the Generalized Bessel–Maitland Function." Discrete Dynamics in Nature and Society 2020 (July 24, 2020): 1–15. http://dx.doi.org/10.1155/2020/1378457.

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In this paper, we aim to determine some results of the generalized Bessel–Maitland function in the field of fractional calculus. Here, some relations of the generalized Bessel–Maitland functions and the Mittag-Leffler functions are considered. We develop Saigo and Riemann–Liouville fractional integral operators by using the generalized Bessel–Maitland function, and results can be seen in the form of Fox–Wright functions. We establish a new operator Zν,η,ρ,γ,w,a+μ,ξ,m,σϕ and its inverse operator Dν,η,ρ,γ,w,a+μ,ξ,m,σϕ, involving the generalized Bessel–Maitland function as its kernel, and also di

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3

Buterin, S. A. "Inverse Spectral Problem for Integrodiﬀerential Sturm-Liouville Operators with Discontinuity Conditions". Contemporary Mathematics. Fundamental Directions 64, № 3 (2018): 427–58. http://dx.doi.org/10.22363/2413-3639-2018-64-3-427-458.

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We consider the Sturm-Liouville operator perturbed by a convolution integral operator on a ﬁnite interval with Dirichlet boundary-value conditions and discontinuity conditions in the middle of the interval. We study the inverse problem of restoration of the convolution term by the spectrum. The problem is reduced to solution of the so-called main nonlinear integral equation with a singularity. To derive and investigate this equations, we do detailed analysis of kernels of transformation operators for the integrodiﬀerential expression under consideration. We prove the global solvability of the

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4

Baleanu, Dumitru, Arran Fernandez, and Ali Akgül. "On a Fractional Operator Combining Proportional and Classical Differintegrals." Mathematics 8, no. 3 (2020): 360. http://dx.doi.org/10.3390/math8030360.

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The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f ( t ) , by a fractional integral operator applied to the derivative f ′ ( t ) . We define a new fractional operator by substituting for this f ′ ( t ) a more general proportional derivative. This new operator can also be written as a Riemann–Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann–Liouville integral and a Caputo derivative

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5

Mayer, F. F., M. G. Tastanov, and A. A. Utemisova. "GEOMETRIC PROPERTIES OF THE BERNATSKY INTEGRAL OPERATOR." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 14, no. 4 (2022): 12–19. http://dx.doi.org/10.14529/mmph220402.

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In the geometric theory of complex variable functions, the study of mapping of classes of regular functions using various operators has now become an independent trend. The connection f(z)∈S o⇔ g(z) = zf'(z) ∈ S * of the classes S o and S * of convex and star-shaped functions can be considered as mapping using the differential operator G[f](x) = zf'(z) of class S o to class S * , that is, G: S o → S * or G(S o ) = S * . The impetus for studying this range of issues was M. Bernatsky's assumption that the inverse operator G –1 [f](x), which translates S * → S o and thereby “improves” the propert

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Tarasov, Vasily E. "Fractional Dynamics with Depreciation and Obsolescence: Equations with Prabhakar Fractional Derivatives." Mathematics 10, no. 9 (2022): 1540. http://dx.doi.org/10.3390/math10091540.

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In economics, depreciation functions (operator kernels) are certain decreasing functions, which are assumed to be equal to unity at zero. Usually, an exponential function is used as a depreciation function. However, exponential functions in operator kernels do not allow simultaneous consideration of memory effects and depreciation effects. In this paper, it is proposed to consider depreciation of a non-exponential type, and simultaneously take into account memory effects by using the Prabhakar fractional derivatives and integrals. Integro-differential operators with the Prabhakar (generalized

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7

Luchko, Yuri. "On a Generic Fractional Derivative Associated with the Riemann–Liouville Fractional Integral." Axioms 13, no. 9 (2024): 604. http://dx.doi.org/10.3390/axioms13090604.

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In this paper, a generic fractional derivative is defined as a set of the linear operators left-inverse to the Riemann–Liouville fractional integral. Then, the theory of the left-invertible operators developed by Przeworska-Rolewicz is applied to deduce its properties. In particular, we characterize its domain, null-space, and projector operator; establish the interrelations between its different realizations; and present a generalized fractional Taylor formula involving the generic fractional derivative. Then, we consider the fractional relaxation equation containing the generic fractional de

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8

Ezquerro, José Antonio, and Miguel Ángel Hernández-Verón. "Using Chebyshev’s polynomials for solving Fredholm integral equations of the second kind." Mathematical Modelling and Analysis 30, no. 1 (2025): 36–51. https://doi.org/10.3846/mma.2025.21036.

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The main problem with the Newton method is the computation of the inverse of the first derivative of the operator involved at each iteration step. Thus, when we want to apply the Newton method directly to solve an integral equation, the existence of the inverse of the first derivative is guaranteed, when the kernel is sufficiently differentiable into any of its two components, through its approximation by Taylor’s polynomial. In this paper, we study the case in which the kernel is not differentiable in any of its two components. So, we present a strategy that consists of approximating the kern

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9

Mika, J., and D. C. Pack. "Approximation to inverses of normal operators." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 103, no. 3-4 (1986): 335–45. http://dx.doi.org/10.1017/s0308210500018989.

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SynopsisFor many purposes, and in particular for the calculation of upper and lower bounds to bilinear forms 〈g0,f〉, where f is the solution to an operator equation and g0 is known, it isuseful to obtain an approximation to the inverse of the operator.For a normal operator A acting in a complex Hilbert space with bounded inverse A−1, we use a direct approach through fundamental results in functional analysis and derive a recipe for the ‘best formula’ for A−1 of form B = βI with β constant and I the identity operator. Examples illustrate that this leads to improved results for certain classes o

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10

Dhaouadi, Lazhar. "Spectral Theory from the Second-Orderq-Difference Operator." International Journal of Mathematics and Mathematical Sciences 2007 (2007): 1–14. http://dx.doi.org/10.1155/2007/16595.

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Spectral theory from the second-orderq-difference operatorΔqis developed. We give an integral representation of its inverse, and the resolvent operator is obtained. As application, we give an analogue of the Poincare inequality. We introduce the Zeta function for the operatorΔqand we formulate some of its properties. In the end, we obtain the spectral measure.

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Boykov, Ilya V., Vladimir A. Roudnev, Alla I. Boykova, and Nikita S. Stepanov. "Continuous operator method application for direct and inverse scattering problems." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 23, no. 3 (2021): 247–72. http://dx.doi.org/10.15507/2079-6900.23.202103.247-272.

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Abstract. We describe the continuous operator method for solution nonlinear operator equations and discuss its application for investigating direct and inverse scattering problems. The continuous operator method is based on the Lyapunov theory stability of solutions of ordinary differential equations systems. It is applicable to operator equations in Banach spaces, including in cases when the Frechet (Gateaux) derivative of a nonlinear operator is irreversible in a neighborhood of the initial value. In this paper, it is applied to the solution of the Dirichlet and Neumann problems for the Helm

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12

Tanirbergenov, M. "INVERSE SCATTERING PROBLEM FOR A SYSTEM OF EQUATIONS DIRAC ON THE WHOLE LINE." Danish scientific journal, no. 71 (April 24, 2023): 36–50. https://doi.org/10.5281/zenodo.7878609.

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<strong>Abstract</strong> In this paper, we study the inverse scattering problem for the Dirac operator on the whole line with real continuous coefficients  and , which tend to zero  rather quickly as and the Dirac operator with these coefficients considered on the semiaxis  has a purely discrete spectrum. For the Dirac operator under consideration, the –function is introduced, its properties are studied, the Gelfand–Levitan–Marchenko integral equation is derived, and the procedure for restoring the coefficients  and  is given.

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13

Barnes, Benedict, C. Sebil, and A. Quaye. "A Generalization of Integral Transform." European Journal of Pure and Applied Mathematics 11, no. 4 (2018): 1130–42. http://dx.doi.org/10.29020/nybg.ejpam.v11i4.3330.

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In this paper, the generalization of integral transform (GIT) of the func-tion G{f (t)} is introduced for solving both differential and interodif-ferential equations. This transform generalizes the integral transformswhich use exponential functions as their kernels and the integral trans-form with polynomial function as a kernel. The generalized integraltransform converts the differential equation in us domain (the trans-formed variables) and reconvert the result by its inverse operator. Inparticular, if u = 1, then the generalized integral transform coincideswith the Laplace transform and thi

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14

GHOSE CHOUDHURY, A., BARUN KHANRA, and A. ROY CHOWDHURY. "CANONICAL BÄCKLUND TRANSFORMATION FOR A DISCRETE INTEGRABLE CHAIN AND ITS ASSOCIATED PROPERTIES." Modern Physics Letters A 18, no. 16 (2003): 1127–39. http://dx.doi.org/10.1142/s0217732303010788.

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The concept of a canonical Bäcklund transformation as laid down by Sklyanin is extended to a discrete integral chain, with a Poisson structure which is not canonical in the strict sense. The transformation is induced by an auxiliary Lax operator with a classical r-matrix which is similar in its algebraic structure to that of the original Lax operator governing the dynamics of the chain. Moreover, the transformation can be obtained from a suitable generating function. It is also shown how successive transformations can be composed to construct a new transformation. Finally an inverse transforma

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15

Murugusundaramoorthy, G., and K. Vijaya. "On Certain Class of Meromorphic Functions with Positive Coefficients Associated with Mittag-Leffler Function Based on Hilbert Space Operator." Владикавказский математический журнал 27, no. 1 (2025): 70–86. https://doi.org/10.46698/p1426-1765-3037-f.

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The Mittag-Leffler~\cite{mit} function ascends naturally in the solution of fractional order differential and integral equations, and exclusively in the studies of fractional generalizing of kinetic equation, random walks, L\'{e}vy flights, super-diffusive transport and in the study of complex systems. In the present investigation, the authors define a new class of meromorphic functions defined in the punctured unit disk $\Delta^*:= \{z\in\mathbb{C}: 0<|z|<1\}$ based on Mittag-Leffler function denoted by $\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)$. We discuss its char

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16

Turmetov, Batirkhan, and Valery Karachik. "On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution." AIMS Mathematics 9, no. 3 (2024): 6832–49. http://dx.doi.org/10.3934/math.2024333.

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<abstract><p>In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two t

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17

Пятков, С. Г., М. В. Уварова, and Т. В. Пронькина. "Inverse problems for a quasilinear parabolic system with integral overdetermination conditions." Журнал «Математические заметки СВФУ», no. 4(108) (December 30, 2020): 43–59. http://dx.doi.org/10.25587/svfu.2020.45.71.004.

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Рассматривается вопрос о корректности в пространствах Соболева обратной задачи определения функции источника для квазилинейной параболической системы второго порядка. Главная часть оператора линейна. В качестве условий переопределения рассматриваются интегральные условия переопределения. Показано, что в случае не более чем линейного роста нелинейного слагаемого по своим аргументам решение существует и единственно в целом по времени и задача корректна в классах Соболева. Условия на данные задачи минимальны. The question of well-posedness in Sobolev spaces of inverse problems of recovering the s

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18

De la Sen, M. "On Bounded Strictly Positive Operators of Closed Range and Some Applications to Asymptotic Hyperstability of Dynamic Systems." Abstract and Applied Analysis 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/639576.

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The problem discussed is the stability of two input-output feedforward and feedback relations, under an integral-type constraint defining an admissible class of feedback controllers. Sufficiency-type conditions are given for the positive, bounded and of closed range feed-forward operator to be strictly positive and then boundedly invertible, with its existing inverse being also a strictly positive operator. The general formalism is first established and the linked to properties of some typical contractive and pseudocontractive mappings while some real-world applications and links of the above

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19

Bolin, David, Kristin Kirchner, and Mihály Kovács. "Numerical solution of fractional elliptic stochastic PDEs with spatial white noise." IMA Journal of Numerical Analysis 40, no. 2 (2018): 1051–73. http://dx.doi.org/10.1093/imanum/dry091.

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Abstract The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in $\mathbb{R}^d$ is considered. The differential operator is given by the fractional power $L^\beta $, $\beta \in (0,1)$ of an integer-order elliptic differential operator $L$ and is therefore nonlocal. Its inverse $L^{-\beta }$ is represented by a Bochner integral from the Dunford–Taylor functional calculus. By applying a quadrature formula to this integral representation the inverse fractional-order operator $L^{-\beta }$ is approximated by a we

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20

A., A. Yusuf. "Subclass of a certain Bazilevic functions associated with Caratheodory functions normalized by other than unity with two radii." Asia Mathematika 5, no. 1 (2021): 76–82. https://doi.org/10.5281/zenodo.4722486.

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Following the approach in \cite{ref6}, we introduced a new subclass of Basilevic function associated with Caratheodory function other than unity define  by a new operator denoted by \(B^{n}_{\sigma, \gamma}(\lambda, \beta)\) and determine the radii of two disks in which the solution of a certain integral equation involving a function \(f \in B^{n}_{\sigma, \gamma}(\lambda, \beta)\) is in \(f \in B^{n}_{\sigma, \gamma}(\lambda, \beta)\) and for which each function \(f \in B^{n}_{\sigma, \gamma}(\lambda, \beta)\) is a \(\sigma-n\)-spiral univalent.

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21

Alsulami, Aishah A., Mariam AL-Mazmumy, Huda O. Bakodah, and Nawal Alzaid. "A Method for the Solution of Coupled System of Emden–Fowler–Type Equations." Symmetry 14, no. 5 (2022): 843. http://dx.doi.org/10.3390/sym14050843.

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A dependable semi-analytical method via the application of a modified Adomian Decomposition Method (ADM) to tackle the coupled system of Emden–Fowler-type equations has been proposed. More precisely, an effective differential operator together with its corresponding inverse is successfully constructed. Moreover, this operator is able to navigate to the closed-form solution easily without resorting to converting the coupled system to a system of Volterra integral equations; as in the case of a well-known reference in the literature. Lastly, the effectiveness of the method is demonstrated on som

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Biondi, Biondo, Sergey Fomel, and Nizar Chemingui. "Azimuth moveout for 3-D prestack imaging." GEOPHYSICS 63, no. 2 (1998): 574–88. http://dx.doi.org/10.1190/1.1444357.

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We introduce a new partial prestack‐migration operator called “azimuth moveout” (AMO) that rotates the azimuth and modifies the offset of 3-D prestack data. Followed by partial stacking, AMO can reduce the computational cost of 3-D prestack imaging. We have successfully applied AMO to the partial stacking of a 3-D marine data set over a range of offsets and azimuths. When AMO is included in the partial‐stacking procedure, high‐frequency steeply dipping energy is better preserved than when conventional partial‐stacking methodologies are used. Because the test data set requires 3-D prestack dept

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23

Beghin, Luisa, and Alessandro De Gregorio. "Stochastic solutions for time-fractional heat equations with complex spatial variables." Fractional Calculus and Applied Analysis 25, no. 1 (2022): 244–66. http://dx.doi.org/10.1007/s13540-021-00011-1.

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AbstractWe deal with complex spatial diffusion equations with time-fractional derivative and study their stochastic solutions. In particular, we complexify the integral operator solution to the heat-type equation where the time derivative is replaced with the convolution-type generalization of the regularized Caputo derivative. We prove that this operator is solution of a complex time-fractional heat equation with complex spatial variable. This approach leads to a wrapped Brownian motion on a circle time-changed by the inverse of the related subordinator. This time-changed Brownian motion is a

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24

Hemeda, A. A. "A Friendly Iterative Technique for Solving Nonlinear Integro-Differential and Systems of Nonlinear Integro-Differential Equations." International Journal of Computational Methods 15, no. 03 (2018): 1850016. http://dx.doi.org/10.1142/s0219876218500160.

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In this work, a simple new iterative technique based on the integral operator, the inverse of the differential operator in the problem under consideration, is introduced to solve nonlinear integro-differential and systems of nonlinear integro-differential equations (IDEs). The introduced technique is simpler and shorter in its computational procedures and time than the other methods. In addition, it does not require discretization, linearization or any restrictive assumption of any form in providing analytical or approximate solution to linear and nonlinear equations. Also, this technique does

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25

Fadili, Ahmed, and Hamid Bounit. "On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems." International Journal of Differential Equations 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/948597.

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This paper studies Volterra integral evolution equations of convolution type from the point of view of complex inversion formula and the admissibility in the Salamon-Weiss sens. We first present results on the validity of the inverse formula of the Laplace transform for the resolvent families associated with scalar Volterra integral equations of convolution type in Banach spaces, which extends and improves the results in Hille and Philllips (1957) and Cioranescu and Lizama (2003, Lemma 5), respectively, including the stronger version for a class of scalar Volterra integrodifferential equations

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Ismailov, M. I., and Cihan Sabaz. "Uniqueness criteria for inverse scattering problem in terms of transmission matrix in boundary condition for a first order system of ordinary differential equations." Baku Mathematical Journal 3, no. 2 (2024): 214–19. http://dx.doi.org/10.32010/j.bmj.2024.18.

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The inverse scattering problem (ISP) involves the recovery of the matrix coefficient of a first-order system on the half-line from its scattering matrix. Specifically, when the matrix coefficient exhibits a triangular structure, the system possesses a Volterra-type integral transformation operator at infinity. This transformation operator facilitates the determination of the scattering matrix on the half-line through matrix Riemann-Hilbert factorization. Solving the ISP on the half-line entails reducing it to an ISP on the whole line for the considered system. This reduction involves extending

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27

Trickovic, Slobodan, and Miomir Stankovic. "On a generalized function-to-sequence transform." Applicable Analysis and Discrete Mathematics, no. 00 (2020): 5. http://dx.doi.org/10.2298/aadm180908005t.

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By attaching a sequence {?n}n?N0 to the binomial transform, a new operator D? is obtained. We use the same sequence to define a new transform T? mapping derivatives to the powers of D?, and integrals to D-1?. The inverse transform B? of T? is introduced and its properties are studied. For ?n = (-1)n, B? reduces to the Borel transform. Applying T? to Bessel's differential operator d/dx x d/dx, we obtain Bessel's discrete operator D?nN?. Its eigenvectors correspond to eigenfunctions of Bessel's differential operator.

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28

Dzhabrailov, Akhmed, Yuri Luchko, and Elina Shishkina. "Two Forms of An Inverse Operator to the Generalized Bessel Potential." Axioms 10, no. 3 (2021): 232. http://dx.doi.org/10.3390/axioms10030232.

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In this paper, we treat a convolution-type operator called the generalized Bessel potential. Our main result is the derivation of two different forms of its inversion. The first inversion is provided in terms of an approximative inverse operator using the method of an improving multiplier. The second one employs the regularization technique for the divergent integrals in the form of the appropriate segments of the Taylor–Delsarte series.

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Pintarelli, Dra María B. "Solution of a system of differential equations with constant coefficients using in-verse moments problem techniques." International Journal of Applied Mathematical Research 7, no. 3 (2019): 71. http://dx.doi.org/10.14419/ijamr.v7i3.12550.

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It is known that given a system of simultaneous linear differential equations with constant coefficients you can apply the Laplace method to solve it. The Laplace transforms are found and the problem is reduced to the resolution of an algebraic system of equations of the determining functions, and applying the inverse transformation the generating functions are determined, solutions of the given system. This implies the need to know the analytical form of the inverse transform of the function. In this case the initial conditions consist in knowing the value that the generating function and its

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Méo, Michel. "A Dual of the Chow Transformation." Complex Manifolds 5, no. 1 (2018): 158–94. http://dx.doi.org/10.1515/coma-2018-0011.

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AbstractWe define a dual of the Chow transformation of currents on the complex projective space. This transformation factorizes a left inverse of the Chow transformation and its composition with the Chow transformation is a right inverse of a linear diferential operator. In such a way we complete the general scheme of integral geometry for the Chow transformation. On another hand we prove the existence of a well defined closed positive conormal current associated to every closed positive current on the projective space. This is a consequence of the existence of a dual current, defined on the d

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Hong-yan, Li, and Wang Xiao-ling. "The generalized inverse of singular integral operators on an open arc and its applications." Wuhan University Journal of Natural Sciences 6, no. 4 (2001): 747–53. http://dx.doi.org/10.1007/bf02850891.

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Choudhuri, Amitava, Benoy Talukdar, and S. B. Dattab. "On a Generalized Fifth-Order Integrable Evolution Equation and its Hierarchy." Zeitschrift für Naturforschung A 61, no. 1-2 (2006): 7–15. http://dx.doi.org/10.1515/zna-2006-1-202.

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A general form of a fifth-order nonlinear evolution equation is considered. The Helmholtz solution of the inverse variational problem is used to derive conditions under which this equation admits an analytic representation. A Lennard type recursion operator is then employed to construct a hierarchy of Lagrangian equations. It is explicitly demonstrated that the constructed system of equations has a Lax representation and two compatible Hamiltonian structures. The homogeneous balance method is used to derive analytic soliton solutions of the third- and fifth-order equations. - PACS numbers: 47.

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Luchko, Yuri. "Some Schemata for Applications of the Integral Transforms of Mathematical Physics." Mathematics 7, no. 3 (2019): 254. http://dx.doi.org/10.3390/math7030254.

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In this survey article, some schemata for applications of the integral transforms of mathematical physics are presented. First, integral transforms of mathematical physics are defined by using the notions of the inverse transforms and generating operators. The convolutions and generating operators of the integral transforms of mathematical physics are closely connected with the integral, differential, and integro-differential equations that can be solved by means of the corresponding integral transforms. Another important technique for applications of the integral transforms is the Mikusinski-

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34

Wazwaz, Abdul-Majid. "A New Integrable Equation Constructed via Combining the Recursion Operator of the Calogero-BogoyavlenskiiSchiff (CBS) Equation and its Inverse Operator." Applied Mathematics & Information Sciences 11, no. 5 (2017): 1241–46. http://dx.doi.org/10.18576/amis/110501.

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Li, Ming, Hai Pu, Lili Cao, et al. "Damage Creep Model of Viscoelastic Rock Based on the Distributed Order Calculus." Applied Sciences 13, no. 7 (2023): 4404. http://dx.doi.org/10.3390/app13074404.

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In this paper, the distributed order calculus was used to establish a creep damage theoretical model to accurately describe the creep properties of viscoelastic materials. Firstly, the definition and basic properties in math of the distributed order calculus were given. On this basis, the mechanical elements of the distributed order damper were built to describe the viscoelastic properties. Then, the distributed order damper was introduced into the three-parameter solid model to establish the distributed order three-parameter solid model. The inverse Laplace transform derived the operator’s co

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Dai, Shikun, Ying Zhang, Kun Li, Qingrui Chen, and Jiaxuan Ling. "Arbitrary Sampling Fourier Transform and Its Applications in Magnetic Field Forward Modeling." Applied Sciences 12, no. 24 (2022): 12706. http://dx.doi.org/10.3390/app122412706.

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Numerical simulation and inversion imaging are essential in geophysics exploration. Fourier transform plays a vital role in geophysical numerical simulation and inversion imaging, especially in solving partial differential equations. This paper proposes an arbitrary sampling Fourier transform algorithm (AS-FT) based on quadratic interpolation of shape function. Its core idea is to discretize the Fourier transform integral into the sum of finite element integrals. The quadratic shape function represents the function change in each element, and then all element integrals are calculated and accum

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Wu, Dan, Yuezan Tao, Jie Yang, and Bo Kang. "Solution to the Unsteady Seepage Model of Phreatic Water with Linear Variation in the Channel Water Level and Its Application." Water 15, no. 15 (2023): 2834. http://dx.doi.org/10.3390/w15152834.

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For a semi-infinite aquifer controlled by a river channel boundary, when the Laplace transform is used to solve a one-dimensional unsteady seepage model of phreatic water while considering the influence of the vertical water exchange intensity ε with the change in the river channel water level f(t), a complicated and tedious integral transformation process is required. By replacing f(t) with an operator, this study first derived the analytic formula of the ε term based on the properties of the Laplace transform without the direct participation of f(t) in the transformation. By using f(t) in th

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Noor, Khhalida Inayat, Bushra Malik, and Syed Zakar Hussain Bukhari. "Some applications of certain integral operators involving functions." Tamkang Journal of Mathematics 49, no. 1 (2018): 25–34. http://dx.doi.org/10.5556/j.tkjm.49.2018.2369.

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Integral transforms map equations from their original domains into others where manipulations and solutions may be much easier than in original domains. To get back in the original environment, we use the idea of inverse of the integral transform. A function analytic and locally univalent in a given simply connected domain is said to be of bounded boundary rotation if its range has bounded boundary rotation which is defined as the total variation of the direction angle of the tangent to the boundary curve under a complete circuit. \qquad The main objective of the present article is to study so

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Nogin, Vladimir A., and Stefan G. Samko. "Method of approximating inverse operators and its applications to the inversion of potential-type integral transforms." Integral Transforms and Special Functions 8, no. 1-2 (1999): 89–104. http://dx.doi.org/10.1080/10652469908819219.

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Kaiafa, Angeliki, and Vassilios Sevroglou. "Interior Elastic Scattering by a Non-Penetrable Partially Coated Obstacle and Its Shape Recovering." Mathematics 9, no. 19 (2021): 2485. http://dx.doi.org/10.3390/math9192485.

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In this paper, the interior elastic direct and inverse scattering problem of time-harmonic waves for a non-penetrable partially coated obstacle placed in a homogeneous and isotropic medium is studied. The scattering problem is formulated via the Navier equation, considering incident circular waves due to point-source fields, where the corresponding scattered data are measured on a closed curve inside the obstacle. Our model, from the mathematical point of view, is described by a mixed boundary value problem in which the scattered field satisfies mixed Dirichlet-Robin boundary conditions on the

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Samraiz, Muhammad, Ahsan Mehmood, Saima Naheed, Gauhar Rahman, Artion Kashuri, and Kamsing Nonlaopon. "On Novel Fractional Operators Involving the Multivariate Mittag–Leffler Function." Mathematics 10, no. 21 (2022): 3991. http://dx.doi.org/10.3390/math10213991.

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The multivariate Mittag–Leffler function is introduced and used to establish fractional calculus operators. It is shown that the fractional derivative and integral operators are bounded. Some fundamental characteristics of the new fractional operators, such as the semi-group and inverse characteristics, are studied. As special cases of these novel fractional operators, several fractional operators that are already well known in the literature are acquired. The generalized Laplace transform of these operators is evaluated. By involving the explored fractional operators, a kinetic differintegral

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Boggess, Albert, та Andrew Raich. "The fundamental solution to \Box_{𝑏} on quadric manifolds – Part 1. General formulas". Proceedings of the American Mathematical Society, Series B 9, № 19 (2022): 186–203. http://dx.doi.org/10.1090/bproc/77.

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This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of C n × C m \mathbb {C}^n\times \mathbb {C}^m . In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator N N and the projection onto the nullspace of ◻ b \Box _b . The main application of our formulas is the critical case of codimension two quadrics in C 4 \mathbb {C}^4 where we discuss the known solvability and hypoellipticity criteria of Peloso and Ricci [J. Funct. A

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HIAI, FUMIO, MILÁN MOSONYI, DÉNES PETZ, and CÉDRIC BÉNY. "QUANTUM f-DIVERGENCES AND ERROR CORRECTION." Reviews in Mathematical Physics 23, no. 07 (2011): 691–747. http://dx.doi.org/10.1142/s0129055x11004412.

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Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f-divergences. Special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of disti

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WAZWAZ, ABDUL-MAJID, RANIA A. ALHARBEY, and S. A. EL-TANTAWY. "A (3+1)-dimensional integrable Calogero-Bogoyavlenskii-Schiff equation and its inverse operator: lump solutions and multiple soliton solutions." Romanian Reports in Physics 75, no. 3 (2023): 116. http://dx.doi.org/10.59277/romrepphys.2023.75.116.

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"In this work, we built a (3+1)-dimensional integrable equation. We started by reformulating the main equation of our model by combining the recursion operator of the Calogero-Bogoyavlenskii-Schiff equation with its inverse recursion op- erator. We confirm the complete integrability of our new developed equation by demon- strating that it satisfies the Painlev´e property. We get a variety of lump solutions that are obtained under specific constraints. Furthermore, we used the simplified Hirota’s direct approach to find multiple soliton solutions to the new evolution equation. In ad- dition, ot

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Melrose, D. B. "Temporal evolution of reactive and resistive nonlinear instabilities." Journal of Plasma Physics 38, no. 3 (1987): 473–81. http://dx.doi.org/10.1017/s0022377800012745.

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A kinetic theory for nonlinear processes involving Langmuir waves, developed in an earlier paper, is extended through consideration of three aspects of the temporal evolution, (i) Following Falk & Tsytovich (1975). the dynamic equation for the rate of change of one amplitude at t is expressed as an integral over T of the product of two amplitudes at t – T and a kernel functionf(T); two generalizations of Falk & Tsytovich's form (f(T) ∝ T) that satisfy the requirement f(∞) = 0 are identified, (ii) It is shown that the low-frequency or beat disturbance may be described in terms of fluctu

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Liu, Yuzhu, Weigang Liu, Zheng Wu, and Jizhong Yang. "Reverse time migration with an exact two-way illumination compensation." GEOPHYSICS 87, no. 2 (2022): S53—S62. http://dx.doi.org/10.1190/geo2020-0815.1.

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Reverse time migration (RTM) has been widely used for imaging complex subsurface structures in oil and gas exploration. However, because only the adjoint of the forward Born modeling operator is applied to the seismic data in RTM, the output migration profile is biased in terms of amplitude. To help partially balance the amplitude performance, the RTM image can be preconditioned with the inverse of the diagonal of the Hessian operator. Yet, existing preconditioning methods do not correctly consider receiver-side effects, assuming that the receiver coverage is infinite or the velocity model is

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CARIÑENA, JOSÉ F., KURUSCH EBRAHIMI-FARD, HÉCTOR FIGUEROA, and JOSÉ M. GRACIA-BOND. "HOPF ALGEBRAS IN DYNAMICAL SYSTEMS THEORY." International Journal of Geometric Methods in Modern Physics 04, no. 04 (2007): 577–646. http://dx.doi.org/10.1142/s0219887807002211.

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The theory of exact and of approximate solutions for non-autonomous linear differential equations forms a wide field with strong ties to physics and applied problems. This paper is meant as a stepping stone for an exploration of this long-established theme, through the tinted glasses of a (Hopf and Rota–Baxter) algebraic point of view. By reviewing, reformulating and strengthening known results, we give evidence for the claim that the use of Hopf algebra allows for a refined analysis of differential equations. We revisit the renowned Campbell–Baker–Hausdorff–Dynkin formula by the modern approa

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Malik, Naseer Ahmad, Farooq Ahmad, and D. K. Jain. "THE THEORETICAL OVERVIEW OF THE HARTLEY TRANSFORM AND THE GENERALIZED R-FUNCTION." Jnanabha 50, no. 01 (2020): 158–63. http://dx.doi.org/10.58250/jnanabha.2020.50116.

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In this paper the R-functions have been mentioned in connection with integral operator named as Hartely transform. The Hartley transform is a mathematical transformation which is closely related to the better known Fourier transform. The properties that differentiate the Hartley Transform from its Fourier counterpart are that the forward and the inverse transforms are identical and also that the Hartley transform of a real signal is a real function of frequency. The Whitened Hartley spectrum, which stems from the Hartley transform, is a bounded function that encapsulates the phase content of a

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Lopushansky, A. O., and H. P. Lopushanska. "Inverse problem for $2b$-order differential equation with a time-fractional derivative." Carpathian Mathematical Publications 11, no. 1 (2019): 107–18. http://dx.doi.org/10.15330/cmp.11.1.107-118.

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We study the inverse problem for a differential equation of order $2b$ with the Riemann-Liouville fractional derivative of order $\beta\in (0,1)$ in time and given Schwartz type distributions in the right-hand sides of the equation and the initial condition. The problem is to find the pair of functions $(u, g)$: a generalized solution $u$ to the Cauchy problem for such equation and the time dependent multiplier $g$ in the right-hand side of the equation. As an additional condition, we use an analog of the integral condition $$(u(\cdot,t),\varphi_0(\cdot))=F(t), \;\;\; t\in [0,T],$$ where the s

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Straub, André. "General formulation of the electric stratified problem with a boundary integral equation." GEOPHYSICS 60, no. 6 (1995): 1656–70. http://dx.doi.org/10.1190/1.1443898.

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The electric potential created by a point source in a stratified model is usually written, in a spectral representation, in terms of a Hankel transform because of the cylindrical symmetry of the model. The solution in the radial wavenumber domain is called the kernel function. This kernel function, as a function of the depth coordinate, is the solution of a 1-D differential equation. The conventional procedure for the calculation of the kernel function consists in applying a recursive scheme. This procedure is effective from a computational point of view but becomes cumbersome from an analytic

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