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Unstable first order time delay systems are frequently encountered in industrial applications, such as chemical plants, hydraulic processes, in satellite communications or economic systems, to name just a few. The control of such processes is a challenging issue. In this paper a filtered Smith Predictor control structure is used to compensate for the process time delays and to ensure the stability of the overall closed loop system. A simplified type of a fractional order PI controller is then designed to meet zero steady state error and an overshoot requirement. The tuning is based on the root locus analysis of the closed loop fractional order system. Simulation examples are provided to validate the proposed method and to demonstrate the efficiency of the proposed control method. Comparisons with two existing methods are included to highlight the possibility of using the proposed method as an alternative solution for controlling these types of processes. [ABSTRACT FROM AUTHOR]

Fractional partial differential equations (FPDEs) have gained significant attention in various scientific and engineering fields due to their ability to describe complex phenomena with memory and long-range interactions. Solving FPDEs analytically can be challenging, leading to a growing need for efficient numerical methods. This review article presents the recent analytical and numerical methods for solving FPDEs, where the fractional derivatives are assumed in Riemann-Liouville’s sense, Caputo’s sense, Atangana-Baleanu’s sense, and others. The primary objective of this study is to provide an overview of numerical techniques commonly used for FPDEs, focusing on appropriate choices of fractional derivatives and initial conditions. This article also briefly illustrates some FPDEs with exact solutions. It highlights various approaches utilized for solving these equations analytically and numerically, considering different fractional derivative concepts. The presented methods aim to expand the scope of analytical and numerical solutions available for time-FPDEs and improve the accuracy and efficiency of the techniques employed.

This paper is to introduce new distributions associated with the linear canonical transform (LCT), which can be regarded as the generalization of the Wigner distribution (WD) and ambiguity function (AF). Moreover, some basic properties of the proposed distributions are also presented. Namely, the shift properties, the conjugation symmetry property, and the marginal properties are proved; the Moyal formula and the anti-derivative property are developed; and the relationship with the short-time Fourier transform (STFT) is also formulated. Furthermore, the newly defined distributions are shown to be valuable and effective in the detection of linear frequency-modulated (LFM) signals. They have better detection performance than the LW, and the LA but are also more flexible than the WDL, and the AFL. [ABSTRACT FROM AUTHOR]

The goal of this article is to develop a spectral collocation method for solving a distributed-order fractional Korteweg-de Vries equation using fractional Zigzag rising diagonal functions. To meet this target, we first introduce Zigzag and Jaiswal polynomials. Then, using a transformation, we find their fractional counterparts. As a linear combination of these functions, we seek a solution to the problem. We will generate operational matrices for fractional Zigzag raising diagonal functions and apply Simpson's rule to approximate the distributed fractional derivative. The resultant approximate equations are collocated to create a system of nonlinear equations. Error analysis of the numerical scheme is fully discussed. Numerical experiments have demonstrated the capability and efficiency of the method. We also demonstrate how the approach might be helpful for problems with non-smooth solutions. [ABSTRACT FROM AUTHOR]

We study a family of representations of the canonical commutation relations (CCR)-algebra, which we refer to as "admissible," with an infinite number of degrees of freedom. We establish a direct correlation between each admissible representation and a corresponding Gaussian stochastic calculus. Moreover, we derive the operators of Malliavin's calculus of variation using an algebraic approach, which differs from the conventional methods. The Fock-vacuum representation leads to a maximal symmetric pair. This duality perspective offers the added advantage of resolving issues related to unbounded operators and dense domains much more easily than with alternative approaches. [ABSTRACT FROM AUTHOR]

In this article, we propose an efficient hybrid method to calculate the highly oscillatory Bessel integral ∫ 0 1 f (x) x - τ J m (ω x γ) d x with the Cauchy type singular point, where 0 < τ < 1 , m ≥ 0 , 2 γ ∈ N +. The hybrid method is established by combining the complex integration method with the Clenshaw– Curtis– Filon– type method. Based on the special transformation of the integrand and the additivity of the integration interval, we convert the integral into three integrals. The explicit formula of the first one is expressed in terms of the Meijer G function. The second is computed by using the complex integration method and the Gauss– Laguerre quadrature rule. For the third, we adopt the Clenshaw– Curtis– Filon– type method to obtain the quadrature formula. In particular, the important recursive relationship of the required modified moments is derived by utilizing the Bessel equation and the properties of Chebyshev polynomials. Importantly, the strict error analysis is performed by a large amount of theoretical analysis. Our proposed methods only require a few nodes and interpolation multiplicities to achieve very high accuracy. Finally, numerical examples are provided to verify the validity of our theoretical analysis and the accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper is to propose a new definition of two-dimensional (2D) Wigner distribution (2D-WD) and two-dimensional ambiguity function (2D-AF) associated with two-dimensional nonseparable linear canonical transform (2D-NS-LCT), namely 2D-NLCWD and 2D-NLCAF. This allows for several consequences of the basic properties of the proposed distributions such as the shift properties, the conjugation symmetry property, the marginal properties, the Moyal formula, and the relationships with the two-dimensional short-time Fourier transform (2D-STFT). Furthermore, we point out the usefulness and efficacy of newly defined distributions for detecting two-dimensional linear frequency-modulated (2D-LFM) signals. [ABSTRACT FROM AUTHOR]

In recent years, the utilization of fractional calculus has witnessed a notable surge across various scientific and engineering domains. This manuscript delves into the exploration of adapted iterative techniques tailored for solving nonlinear equations, capitalizing on the diverse range of derivatives available for addressing different problem contexts. We scrutinize previously developed iterative methods, enhancing their efficacy by introducing an auxiliary parameter to the root search process for nonlinear equations (NLE), alongside a fixed order of fractional derivatives. The selection of the auxiliary parameter is confined to the interval (0,1] for convenience. A thorough convergence analysis is conducted, employing a fractional power series expansion of f(x) in terms of fractional derivatives. Subsequently, a series of NLEs is solved to showcase and contrast the efficiency of our proposed methods with established iterative techniques. This refined abstract aims to succinctly elucidate the objectives and contributions of our study, providing readers with a clearer understanding of the manuscript's scope and significance.

This paper is devoted to explore the convolution quadrature based on a class of two-point Hermite collocation methods. Incorporating derivatives into the numerical scheme enhances the accuracy while preserving stability, which is confirmed by the convergence analysis for the discretization of the initial value problem. Moreover, we employ the resulting quadrature to evaluate a class of highly oscillatory integrals. The frequency-explicit convergence analysis demonstrates that the proposed convolution quadrature surpasses existing convolution quadratures, achieving the highest convergence rate with respect to the oscillation among them. Numerical experiments involving convolution integrals with smooth, weakly singular, and highly oscillatory Bessel kernels illustrate the reliability and efficiency of the proposed convolution quadrature. [ABSTRACT FROM AUTHOR]

We study the computation of the zero set of the Bargmann transform of a signal contaminated with complex white noise, or, equivalently, the computation of the zeros of its short-time Fourier transform with Gaussian window. We introduce the adaptive minimal grid neighbors algorithm (AMN), a variant of a method that has recently appeared in the signal processing literature, and prove that with high probability it computes the desired zero set. More precisely, given samples of the Bargmann transform of a signal on a finite grid with spacing δ , AMN is shown to compute the desired zero set up to a factor of δ in the Wasserstein error metric, with failure probability O (δ 4 log 2 (1 / δ)) . We also provide numerical tests and comparison with other algorithms. [ABSTRACT FROM AUTHOR]

We consider a problem of dynamic 2D vector tomography, i.e. the object under investigation changes during the data acquisition. More precisely, we consider the case when the object motion is a combination of rotation and shifting. The task is then to reconstruct the searched-for vector field by known values of the dynamic ray transforms. In order to solve this dynamic inverse problem, we first study properties of the dynamic ray transforms operators. In particular, the singular value decompositions of the operators are constructed using classic orthogonal polynomials. Following from this study, a numerical algorithm for solving the dynamic problem is proposed based on the truncated singular value decomposition method. [ABSTRACT FROM AUTHOR]

The ambiguity function (AF) and Wigner distribution (WD) play an important role not only in non-stationary signal processing but also in radar and sonar systems. In this paper, we introduce modified ambiguity function and Wigner distribution associated with quadratic-phase Fourier transform (QAF, QWD). Moreover, many various useful properties of QAF and QWD are also proposed. Marginal properties and Moyal’s formulas of these distributions have elegance and simplicity comparable to those of the AF and WD. Besides, convolutions via quadratic-phase Fourier transform are also introduced. Furthermore, convolution theorems for QAF and QWD are also derived, which seem similar to those of the classical Fourier transform (FT). In addition, applications of QAF and QWD are established such as the detection of the parameters of single-component and multi-component linear frequency-modulated (LFM) signals. [ABSTRACT FROM AUTHOR]

The main goal of this study was to look into some new integral transformations that are associated with a generalized kk-Bessel function. Integral formulas for the generalized kk-Bessel function have been established using the Laplace transform, Euler transform, Whittaker transform, and kk-transforms. The results presented here have the potential to be helpful, and some special cases of corollaries are explicitly demonstrated.

We study the problem of the integral geometry, in which the functions are integrated over hyperplanes in the n-dimensional Euclidean space, n = 2 ⁢ m + 1 . The integrand is the product of a function of n variables called the density and weight function depending on 2 ⁢ n variables. Such an integration is called here the weighted Radon transform, which coincides with the classical one if the weight function is equal to one. It is proved the uniqueness for the problem of determination of the surface on which the integrand is discontinuous. [ABSTRACT FROM AUTHOR]

In this paper, the He–Elzaki transform method (HEM) is proposed. The method is formulated by combining He's variation iteration method and the modified Laplace transform, known as the Elzaki integral transform. This method is designed to solve the time-fractional telegraph equation that arises in electromagnetics. The Caputo sense is used to describe fractional derivatives. One of the advantages of this method is that the computation of the Lagrange multiplier is not necessarily required through the convolution theorem or integration in recurrence relations. Additionally, to reduce nonlinear computational terms, He's polynomial is determined using the homotopy perturbation method. The proposed method is applied to several examples of nonlinear fractional telegraph equations. The results obtained from these examples demonstrate that the proposed method is an efficient technique that facilitates the process of solving time-fractional differential equations. [ABSTRACT FROM AUTHOR]

Nonlocal operators with different kernels were used here to obtain more general harmonic oscillator models. Power law, exponential decay, and the generalized Mittag-Leffler kernels with Delta-Dirac property have been utilized in this process. The aim of this study was to introduce into the damped harmonic oscillator model nonlocalities associated with these mentioned kernels and see the effect of each one of them when computing the Bode diagram obtained from the Laplace and the Sumudu transform. For each case, we applied both the Laplace and the Sumudu transform to obtain a solution in a complex space. For each case, we obtained the Bode diagram and the phase diagram for different values of fractional orders. We presented a detailed analysis of uniqueness and an exact solution and used numerical approximation to obtain a numerical solution.

In this article, an approximation of the image of the closed ball of the space Lp{L}_{p} (p>1p\gt 1) centered at the origin with radius rr under Hilbert-Schmidt integral operator F(⋅):Lp→LqF\left(\cdot ):{L}_{p}\to {L}_{q}, 1p+1q=1\frac{1}{p}+\frac{1}{q}=1 is considered. An error evaluation for the given approximation is obtained.

In this article, we study the semi discrete and fully discrete formulations for a Kirchhoff type quasilinear integro-differential equation ( D α ) involving time-fractional derivative of order α ∈ (0 , 1) . For the semi discrete formulation of the equation ( D α ), we discretize the space domain using a conforming FEM and keep the time variable continuous. We modify the standard Ritz–Volterra projection operator to carry out error analysis for the semi discrete formulation of the considered equation. In general, solutions of the time-fractional partial differential equations have a weak singularity near time t = 0 . Taking this singularity into account, we develop a new linearized fully discrete numerical scheme for the equation ( D α ) on a graded mesh in time. We derive a priori bounds on the solution of this fully discrete numerical scheme using a new weighted H 1 (Ω) norm. We prove that the developed numerical scheme has an accuracy rate of O (P - 1 + N - (2 - α)) in L ∞ (0 , T ; L 2 (Ω)) as well as in L ∞ (0 , T ; H 0 1 (Ω)) , where P and N are degrees of freedom in the space and time directions respectively. The robustness and efficiency of the proposed numerical scheme are demonstrated by some numerical examples. [ABSTRACT FROM AUTHOR]

We present a fast and approximate multifrontal solver for large-scale sparse linear systems arising from finite-difference, finite-volume, or finite-element discretization of high-frequency wave equations. The proposed solver leverages the butterfly algorithm and its hierarchical matrix extension for compressing and factorizing large frontal matrices via graph distance guided entry evaluation or randomized matrix-vector multiplication-based schemes. Complexity analysis and numerical experiments demonstrate O(N log2 N) computation and O(N) memory complexity when applied to an N × N sparse system arising from 3D high-frequency Helmholtz and Maxwell problems.

We study the X-ray transform over a generic family of smooth curves in R 2 with a Riemannian metric g. We show that the singularities cannot be recovered from local data in the presence of conjugate points, and therefore artifacts may arise in the reconstruction. We perform numerical experiments to illustrate the results. [ABSTRACT FROM AUTHOR]

We propose a generalized method for solving the Gelfand–Levitan–Marchenko equation (GLME) based on the block version of the Toeplitz Inner-Bordering (TIB). The method works for the signals containing both the continuous and the discrete spectra. The method allows us to calculate the potential at an arbitrary point and does not require small spectral data. Using this property, we can perform calculations to the right and to the left of the selected starting point. For the discrete spectrum, the procedure of cutting off exponentially growing matrix elements is suggested to avoid the numerical instability and perform calculations for soliton solutions spaced apart in the time domain. [ABSTRACT FROM AUTHOR]

We present quadrature schemes to calculate matrices where the so-called modified Hilbert transformation is involved. These matrices occur as temporal parts of Galerkin finite element discretizations of parabolic or hyperbolic problems when the modified Hilbert transformation is used for the variational setting. This work provides the calculation of these matrices to machine precision for arbitrary polynomial degrees and non-uniform meshes. The proposed quadrature schemes are based on weakly singular integral representations of the modified Hilbert transformation. First, these weakly singular integral representations of the modified Hilbert transformation are proven. Second, using these integral representations, we derive quadrature schemes, which treat the occurring singularities appropriately. Thus, exponential convergence with respect to the number of quadrature nodes for the proposed quadrature schemes is achieved. Numerical results, where this exponential convergence is observed, conclude this work. [ABSTRACT FROM AUTHOR]

We present a fast and approximate multifrontal solver for large-scale sparse linear systems arising from finite-difference, finite-volume, or finite-element discretization of high-frequency wave equations. The proposed solver leverages the butterfly algorithm and its hierarchical matrix extension for compressing and factorizing large frontal matrices via graph distance guided entry evaluation or randomized matrix-vector multiplication-based schemes. Complexity analysis and numerical experiments demonstrate O(N log2 N) computation and O(N) memory complexity when applied to an N × N sparse system arising from 3D high-frequency Helmholtz and Maxwell problems.

In this article, we introduce a novel scaling Wigner distribution by intertwining the merits of fractional instantaneous auto-correlation and the linear canonical transform. We initiate the study by investigating the fundamental properties of the scaling Wigner distribution, including the nonlinearity, marginal, shifting, conjugate-symmetry, convolution, and anti-derivative properties. Subsequently, the Moyal's formula is also examined in detail. To testify the efficiency, the proposed distribution is employed for the detection of single-component, bi-component, and tri-component linear frequency modulated signals. The simulation results lucidly demonstrate that the scaling Wigner distribution performs exceptionally well in comparison with the conventional Wigner distribution. Nonetheless, owing to the higher degrees of freedom of the linear canonical transform, the proposed distribution benefits in cross-term reduction while maintaining a perfect time–frequency resolution and clear auto-terms angle resolution. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new source condition and introduce a new apriori parameter choice strategy for Lavrentiev regularization method for nonlinear ill-posed operator equation involving a monotone operator in the setting of a Hilbert space. Also, a fifth order iterative method is being proposed for approximately solving Lavrentiev regularized equation. A numerical example is illustrated to demonstrate the performance of the method. [ABSTRACT FROM AUTHOR]

Let R denote the generalized Radon transform, which integrates over a family of N-dimensional smooth submanifolds S y ~ ⊂ U , 1 ≤ N ≤ n - 1 , where an open set U ⊂ R n is the image domain. The submanifolds are parametrized by points y ~ ⊂ V ~ , where an open set V ~ ⊂ R n is the data domain. We assume that the canonical relation C ~ from T ∗ U to T ∗ V ~ of R is a local canonical graph (when R is viewed as a Fourier Integral Operator). The continuous data are denoted by g, and the reconstruction is f ˇ = R ∗ B g . Here R ∗ is a weighted adjoint of R , B is a pseudo-differential operator, and g is a conormal distribution. Discrete data consists of the values of g on a regular lattice with step size O (ϵ) . Let S denote the singular support of f ˇ , and f ˇ ϵ = R ∗ B g ϵ be the reconstruction from interpolated discrete data g ϵ (y ~) . Pick a point x 0 ∈ S , i.e. the singularity of f ˇ at x 0 is visible from the data. The main result of the paper is the computation of the limit DTB (x ˇ) : = lim ϵ → 0 ϵ κ f ˇ ϵ (x 0 + ϵ x ˇ). Here κ ≥ 0 is selected based on the strength of the reconstructed singularity, and x ˇ is confined to a bounded set. The limiting function DTB (x ˇ) , which we call the discrete transition behavior, contains full information about the resolution of reconstruction. [ABSTRACT FROM AUTHOR]

Our recent work (Kang and Wang in J Sci Comput 82:1–33, 2020) performed a complete asymptotic analysis and proposed a modified Filon-type method for a class of oscillatory infinite Bessel transform with a general oscillator. In this paper, we present and analyze a different method by converting the integration path to the complex plane for this class of oscillatory infinite Bessel transform. In particular, we establish a series of new quadrature rules for this transform and carry out rigorous analysis, including the cases that the oscillator g(x) has either zeros or stationary points. The error analysis shows the advantages that this approach exhibits high asymptotic order, and the accuracy improves significantly as either the frequency ω or the number of nodes n increases. Furthermore, the constructed method shows higher accuracy and error order by comparing with the existing modified Filon-type method in our recent work (Kang and Wang 2020) at the same computational cost. Some numerical experiments are provided to verify the theoretical results and demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The aim of this study is to introduce a convolution-based two-dimensional fractional shearlet transform in the context of fractional time-frequency analysis. The preliminary analysis encompasses the derivation of fundamental properties of the novel integral transform including the orthogonality relation, inversion formula, and the range theorem. To extend the scope of the study, the cone adapted variant of the two-dimensional fractional shearlet transform is also studied in detail. Nevertheless, several coherent examples are presented to facilitate a sound illustration of the concepts. [ABSTRACT FROM AUTHOR]

The ripplet transform is a refinement of the ordinary curvelet transform based on the non-parabolic scaling law for resolving two-dimensional singularities. However, the ripplet transform offers flexibility only in the angular window which is not feasible for optimizing the concentration of the ripplet spectrum. Therefore, there arises a fundamental question “whether it is possible to induce flexibility in both the radial and angular windows of the ordinary ripplet transform”. The aim of this study is to address this issue by constructing a new family of ripplet waveforms based on the linear canonical transform having certain extra degrees of freedom for resolving higher dimensional singularities. To begin with, we examine the fundamental aspects of the novel ripplet transform including the formulation of reconstruction and Rayleigh’s energy formulae. In the sequel, we obtain a Heisenberg-type uncertainty inequality associated with the novel ripplet transform. Towards the culmination, we introduce and study a new class of localization operators associated with the novel ripplet transform. [ABSTRACT FROM AUTHOR]

In this paper, we study the fundamental properties of the Hartley convolution operator and its applications, which specifically are the normed ring structures on the L 1 (R) space when being equipped with the multiplication defined as Hartley convolution multiplication. We prove the Young-type theorem and Young's inequalities for the Hartley convolution operator in the one-dimensional case. Furthermore, we also formulate the Watson-type transformation for the Hartley convolution operator. We establish necessary and sufficient conditions for this operator to be unitary on L 2 (R) and get its inverse represented in the conjugate symmetric form. In addition to showing properties of this operator, we use the obtained results to study the solvability in close-form and estimate the boundedness solutions of some classes for integro-differential equations of Barbashin type and the Fredholm-type integral equation. Finally, we build numerical examples for illustrating the obtained results to ensure their validity and applicability. [ABSTRACT FROM AUTHOR]

In this work, we construct two numerical approximations methods to deal with approximations of a linear Fredholm integro-differential equation. In order to show the existence and uniqueness of the solution we start by the reformulation of the integro-differential equation to a system of integral equations. We explain the general framework of the projection method which helps to clarify the basic ideas of the Collocation and Kantorovich methods. We introduce a new iterative method to avoid inversing the block operator matrix. Next, we apply the iterative projection methods and we present theorems to show the convergence of the constructed solutions to the exact solution. Finally, to observe the error behavior of the two methods, we give two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, the modified integral equation, namely, Elzaki transformation coupled with the Adomian decomposition method called Elzaki Adomian decomposition method (EADM) is used to investigate the solution of time-fractional fourth-order parabolic partial differential equations (PDEs) with variable coefficients. The introduced method is used to solve two models of the proposed problem, the analytical and approximate solutions of the models are obtained. The outcomes illustrate that the proposed technique is a highly accurate, and facilitates the process of solving differential equations by comparing it, with the exact solution and those obtained by the variation iteration method (VIM) and Laplace homotopy perturbation method (LHPM). [ABSTRACT FROM AUTHOR]

In this article, we create an effective technique fundamental on 2-dimensional Chelyshkov polynomials (2D-CPs) to expand an estimate solution of a type of 2-dimensional nonlinear Volterra integral equations (2D-NVIEs) of the primary type. This problem is changed to a 2D-NVIE of the 2 type and the technique transforms the integral equation to a system of nonlinear algebraic equations via implies of collocation points. At last, some numerical examples are given to clarify the efficiency of the presented strategy. [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse problem of the generalized X-ray transform in n-dimensional Schwartz space. We extend Natterer's inversion formula to the generalized X-ray transform. Moreover, we extend results of Rigaud and Lakhal to the n-dimensional generalized X-ray transform. Furthermore, we gain some Sobolev estimate formulas of the generalized X-ray transform. [ABSTRACT FROM AUTHOR]

This paper proposes a multidimensional Hilbert transform approach for pricing discretely monitored multi-asset barrier options and computing joint survival probability in multivariate exponential Lévy asset price models. We generalize the univariate Hilbert transform method of Feng and Linetsky (Math Financ 18(3), 337–384, 2008) for single-asset barrier options and the well-known Sinc approximation theory of Stenger (Numerical methods based on sinc and analytic functions. Springer, New York, 1993) for computing the one-dimensional Hilbert transform to any dimension. We prove that, for Lévy processes with joint characteristic functions having an exponentially decaying tail, the error of our method decays exponentially in some power of the number of terms used in the expansion for each dimension. Numerical experiments demonstrate the efficiency of our method in the two-dimensional and three-dimensional problems for some popular multivariate Lévy models. [ABSTRACT FROM AUTHOR]

This paper presents a novel and elegant convolution structure for the multi-dimensional linear canonical transform involving a pure multi-dimensional kernel obtained via a general 2 n × 2 n real, symplectic matrix M with n (2 n + 1) independent parameters. The primary intention is to develop the convolution theorem associated with the novel linear canonical convolution. The convolution structure is subsequently invoked to establish the sampling theorem for the band-limited signals in the multi-dimensional linear canonical domain. The validity and efficiency of the sampling procedure are demonstrated via a lucid example. Besides, the Heisenberg's and Beckner's uncertainty principles associated with the multi-dimensional linear canonical transform are also studied in detail. Finally, we study and design the multiplicative filter in the multi-dimensional linear canonical domain by utilizing the proposed multi-dimensional convolution structure. [ABSTRACT FROM AUTHOR]

In the present paper, we propose a spectral collocation method based on Pell polynomials to obtain the solution of a variable-order fractional integro-differential equation with a weakly singular kernel. Fractional integro-differential equations are used in many mathematical models, including heat conduction in memory materials, nuclear reactor dynamics, and chemical kinetics. To provide a numerical scheme, we consider Pell polynomials and use operational matrices to approximate variable-order Caputo fractional derivative as well as integral terms in the main equation. A linear system of equations is formed by collocating the resulting approximate equations. The existence and uniqueness of the solution to the main equation have been proved. The convergence of the approximation solution has been discussed. Several test problems are reported to demonstrate the validity of the method. [ABSTRACT FROM AUTHOR]

An attempt is made here to study the Mittag–Leffler function with two variables. Its various properties including integral and operational relationships with other known Mittag–Leffler functions of one variable, pure and differential recurrence relations, Euler transform, Laplace transform, Mellin transform, Whittaker transform, Mellin–Barnes integral representation, and its relationship with Wright hypergeometric function are investigated and established. Also, properties of the Mittag–Leffler function of two variables associated with fractional calculus operators are considered. [ABSTRACT FROM AUTHOR]

This paper is concerned with the use of the focal underdetermined system solver to recover sparse empirical quadrature rules for parametrized integrals from existing data. This algorithm, originally proposed for image and signal reconstruction, relies on an approximated ℓ p {\ell^{p}} -quasi-norm minimization. Compared to ℓ 1 {\ell^{1}} -norm minimization, the choice of 0 < p < 1 {0

The integrals of arbitrary products of zero-order and first-order Bessel functions (double Bessel functions) are widely used in practical problems. However, due to the strong oscillation and slow decay properties of the product terms of the double Bessel function, traditional numerical integration algorithms are no longer applicable. It is particularly important to develop high-precision and high-efficiency numerical algorithms for these types of integrals. In this paper, using the large argument approximate expression of the Bessel function and the accumulation property of the integral, double Bessel function integrals are transformed into the algebraic sum of ordinary integral forms in finite region and Fourier transform forms. The folded linear approach method and the Gaver–Stehfest (G–S) inverse Laplace transform method are developed for double Bessel function integrals and double-order single-argument Hankel transforms. Numerical examples are given to verify the efficiency and precision of these methods. The results show that the precision of the folded linear approach method is higher than that of the G–S inverse Laplace transform method, while the latter is more efficient and runs several times faster than the former. These two methods have different requirements for F (λ) when using the double Bessel function integrals. Therefore, the method that should be selected depends on the requirements of the specific F (λ) problem. This paper presents reliable computing methods for the study of fracture mechanics, electromagnetic responses, acoustic wave scattering, and droplet wetting. [ABSTRACT FROM AUTHOR]

The Compton camera is a gamma ray imaging device expected to provide clinically relevant images in the SPECT applications where collimated cameras are sub-optimal. Its imaging performances depend not only on the design of the detection system but also on choices related to tomographic reconstruction. The aim of this work is to show that the accuracy in modelling the acquisition largely influences the quality of the images. For this purpose, we restrict here to Doppler broadening models in conjunction with the list-mode maximum likelihood expectation maximization (LM-MLEM) algorithm. The study was carried out with Monte-Carlo simulation. We show that the reconstructed point spread function is location-dependent when the model is not accurate, and the usual elongation artefacts well-known in Compton camera imaging will appear. The model we propose allows us to reconstruct isolated point sources and more complex non-uniform sources with improved resolution even in the direction orthogonal to the camera. [ABSTRACT FROM AUTHOR]