Abel-type Integral Equations Research Articles

Generalizations of Riemann–Liouville fractional integrals and applications

The notion of a generalized Riemann–Liouville fractional integral is introduced, and its domain, range, and properties are studied. The new notion and properties provide new insight and understanding into the classical Riemann–Liouville fractional integral and its properties. Based on the generalized Riemann–Liouville fractional integral, equivalences between linear first ‐order fractional differential equations (FDEs) and integral equations are established. These equivalence results are applied to obtain solutions of Abel type integral equations and to study the existence and uniqueness of generalized normal solutions of initial value problems for nonlinear first‐order FDEs.

An efficient numerical scheme to solve generalized Abel’s integral equations with delay arguments utilizing locally supported RBFs

Hereditary effects are commonly observed in diverse scientific domains such as engineering, economics, biology, mathematics, and physics. In the model of atomic irradiation of solids with unbounded cross-sectional areas, determining the average number of atoms displaced has been achieved through delay systems that incorporate the consideration of past states. In this study, we employ the discrete collocation method using local radial basis functions to numerically solve Abel-type integral equations with a delay argument. This method balances accuracy, efficiency, and flexibility and is well-suited for complex practical problems as it requires less memory and computational volume compared to its global counterpart. Instead of approximating the solution at all points of the domain, the method considers a set of nodes in the neighborhood of a certain point. As a result, the method can be readily implemented on a standard personal computer without requiring high-end specifications. To calculate the singular integrals the nonuniform composite Gauss–Legendre numerical integration rule is employed. We discuss the error analysis and convergence rate of the offered scheme and test it with several numerical examples. The results obtained are also consistent with the theoretical error analysis expectations.

A system of Abel type integral equations

A system of Abel type integral equations

A novel numerical approach based on shifted second‐kind Chebyshev polynomials for solving stochastic Itô–Volterra integral equation of Abel type with weakly singular kernel

In this paper, a collocation method based on shifted second‐order Chebyshev polynomials is implemented to obtain the approximate solution of the stochastic Itô–Volterra integral equation of Abel type with weakly singular kernel. In this method, operational matrices are used to convert the stochastic Itô–Volterra integral equation to algebraic equations that are linear. The algorithm of the proposed numerical scheme has been presented in this paper. Also, the error bound and convergence of the proposed method are well established. Consequently, two illustrative examples are provided to demonstrate the efficiency, plausibility, reliability, and consistency of the current methodology.

Stability Domains of an Implicit Method for the Numerical Solution of Abel Type Integral Algebraic Equations

Stability Domains of an Implicit Method for the Numerical Solution of Abel Type Integral Algebraic Equations

Stability domains of an implicit method for the numerical solution of Abel type integral algebraic equations

Stability domains of an implicit method for the numerical solution of Abel type integral algebraic equations

Non-Fourier thermal shock resistance and transient thermal fracture of magneto-electro-elastic composite with a penny-shaped crack

Magneto-electro-elastic composite with a penny-shaped crack is analyzed. The exact solution of temperature field based on non-Fourier heat conduction is derived by the method of standard separation of variables. The thermal stress, electrical displacement and magnetic induction in the transversely isotropic andaxis-symmetric magneto-electro-elasticcylinder are evaluated by Hankel transform technique. The thermal stress, electrical displacement and magnetic induction intensity factors at the tip of penny-shaped crack are obtained by Abel type integral equation. The influence of the thermal relaxation time, crack length, characteristic length of materials and magneto-electro-thermo-elastic coupling on intensity factors are analyzed. Thermal shock resistance of MEE cylinder is evaluated by associating the stress based failure criterion and fracture mechanics based failure criterion.

Remarks on coupled Abel type integral equations

Remarks on coupled Abel type integral equations

Asteroseismology as a new window to statistics of binaries

More than 340 non-eclipsing binaries of A-F stars as primaries at intermediate periods (100 - 1000 d) were newly found by uninterrupted photometry with ultra high-precision taken over 4 yr by Kepler space mission via the phase modulation of pulsating stars, and their orbital parameters, which were difficult to measure with conventional methods and were very incomplete, were then determined. This asteroseismic finding of binaries tripled the number of intermediate-mass binaries with full orbital solutions, and opened a new window to statistics of binaries. We outline the methods of finding non-eclipsing binaries by photometry and of deriving the orbital parameters, and demonstrate their validation. Statistical study of the newly discovered binaries along with the known spectroscopic binaries near A-F main-sequence are then presented, converting distribution in the mass-function into the mass-ratio distribution with the help of Abel-type integral equation.

Exact and Numerical Solution of Abel Integral Equations by Orthonormal Bernoulli Polynomials

In this study, we use the Bernoulli polynomials, Bernoulli number to construct the orthonormal polynomials by using Gram–Schmidt orthogonalization. We use the function approximation on orthonormal polynomials, apply the integration operator on these orthonormal polynomials and obtain tri-diagonal operational matrix of integration. Apply the tri-diagonal operational matrix on many Abel-type integral equations and change all integral equations to the algebraic equation solutions. The exact and approximate solution of Abel-type integral equations by this new method will be found out. To illustrate the applicability, efficiency and accuracy of suggested scheme, some numerical examples of Abel-type integral equations are implemented and the comparisons are given by exact solutions.

Orthonormal Bernoulli polynomials collocation approach for solving stochastic Itô‐Volterra integral equations of Abel type

AbstractIn this paper, orthonormal Bernoulli collocation method has been developed to obtain the approximate solution of linear singular stochastic Itô‐Volterra integral equations. By applying this method, linear stochastic integral equation converts to linear system of algebraic equations. This system is achieved by approximating functions that appear in the stochastic integral equations by using orthonormal Bernoulli polynomials (OBPs) and then substituting these approximations into consideration equation. This linear system of algebraic equations can be solved via an appropriate numerical method and approximate solution of integral equation is obtained. A main advantage of this technique is that the condition number of the coefficient matrix of the system is small, which verify that THE proposed method is stable. Also, convergence and error analysis of the present method are discussed. Finally, two examples are given to show the pertinent properties, applicability, and accuracy of the present method.

Recovering functions from the spherical mean transform with limited radii data by expansion into spherical harmonics

The aim of the article is to generalize the method presented in [3, Theorem 1] by G. Ambartsoumian, R. Gouia-Zarrad and M. Lewis for recovering functions from their spherical mean transform with limited radii data from the two dimensional case to the general n dimensional case. The idea behind the method is to expand each function in question into spherical harmonics and then obtain, for each term in the expansion, an integral equation of Volterra's type that can be solved iteratively. We show also how this method can be modified for the spherical case of recovering functions from the spherical transform with limited radii data. Lastly, we solve the analogous problem for the case of the Funk transform by again using expansion into spherical harmonics and then obtain an Abel type integral equation which can be inverted by a method introduced in [14].

Generalized Abel type integral equations with localized fractional integrals and derivatives

ABSTRACTGeneralized Abel type integral equations with Gauss, Kummer's and Humbert's confluent hypergeometric functions in the kernel and generalized Abel type integral equations with localized fractional integrals are considered. The left-hand sides of these equations are inversed by using generalized fractional derivatives. Explicit solutions of the equations in the class of locally summable functions are obtained. They are represented in terms of hypergeometric functions. Asymptotic power exponential type expansions of the generalized and localized fractional integrals are obtained. The base solutions of the generalized Abel type integral equation are given in the form of asymptotic series.

A fully discrete Galerkin method for Abel-type integral equations

In this paper, we present a Galerkin method for Abel-type integral equation with a general class of kernel. Stability and quasi-optimal convergence estimates are derived in fractional-order Sobolev norms. The fully-discrete Galerkin method is defined by employing simple tensor-Gauss quadrature. We develop a corresponding perturbation analysis which allows to keep the number of quadrature points small. Numerical experiments have been performed which illustrate the sharpness of the theoretical estimates and the sensitivity of the solution with respect to some parameters in the equation.

Electric and magnetic polarization saturations for a thermally loaded penny-shaped crack in a magneto-electro-thermo-elastic medium

This paper is devoted to investigating the thermal-induced electric and magnetic polarization saturations (PS) at the tip of a penny-shaped crack embedded in an infinite space of magneto-electro-thermo-elastic medium. In view of the symmetry with respect to the cracked plane, this crack problem is formulated by a mixed boundary value problem. By virtue of the solution to the Abel type integral equation, the governing equations corresponding to the present problem are analytically solved and the generalized crack surface displacement and field intensity factors are obtained in closed-forms. Applying the hypothesis of the electric and magnetic PS model to the analytical results, the sizes of the electric and magnetic yielding zones are determined. Numerical calculations are carried out to reveal the influences of the thermal load and the electric and magnetic yielding strengths on the results, and to show the distributions of the electric and magnetic potentials on the crack surfaces. It is found that the sizes of electric and magnetic yielding zones are mainly dependent on the electric and magnetic yielding strengths, respectively. Since the multi-ferroic media are widely used in various complex thermal environments, the present work could serve as a reference for the designs of various magneto-electric composite structures.

Applications of fractional calculus in solving Abel-type integral equations: Surface–volume reaction problem

In this paper we consider a class of partial integro-differential equations of fractional order, motivated by an equation which arises as a result of modeling surface–volume reactions in optical biosensors. We solve these equations by employing techniques from fractional calculus; several examples are discussed. Furthermore, for the first time, we encounter an order of the fractional derivative other than 12 in an applied problem. Hence, in this paper we explore the applicability of fractional calculus in real-world applications, further strengthening the true nature of fractional calculus.

A spectral iterative method for solving nonlinear singular Volterra integral equations of Abel type

In this paper, a spectral iterative method is employed to obtain approximate solutions of singular nonlinear Volterra integral equations, called Abel type of Volterra integral equations. The Abel's type nonlinear Volterra integral equations are reduced to nonlinear fractional differential equations. This approach is based on a combination of two different methods, i.e. the iterative method proposed in [7] and the spectral method. The method reduces the fractional differential equations to systems of linear algebraic equations and then the resulting systems are solved by a numerical method. Finally, we prove that the spectral iterative method (SIM) is convergent. Numerical results comparing this iterative approach with alternative approaches offered in [4,8,24] are presented. Error estimation also corroborate numerically.

A New Algorithm for a Nonlinear Abel Type Integral Equation That Determines the Temperature in a Semi-Infinite Solid

A New Algorithm for a Nonlinear Abel Type Integral Equation That Determines the Temperature in a Semi-Infinite Solid

Parametric Iteration Method and Weakly Singular Volterra Integral Equations of Abel Type

Parametric Iteration Method and Weakly Singular Volterra Integral Equations of Abel Type

Nonparametric maximum likelihood estimation for the multisample Wicksell corpuscle problem.

We study nonparametric maximum likelihood estimation for the distribution of spherical radii using samples containing a mixture of one-dimensional, two-dimensional biased and three-dimensional unbiased observations. Since direct maximization of the likelihood function is intractable, we propose an expectation-maximization algorithm for implementing the estimator, which handles an indirect measurement problem and a sampling bias problem separately in the E- and M-steps, and circumvents the need to solve an Abel-type integral equation, which creates numerical instability in the one-sample problem. Extensions to ellipsoids are studied and connections to multiplicative censoring are discussed.