Two classes of conformable fractional Sturm‐Liouville problems: Theory and applications

In this paper, we investigate in more detail some useful theorems related to conformable fractional derivative (CFD) and integral and introduce two classes of conformable fractional Sturm‐Liouville problems (CFSLPs): namely, regular and singular CFSLPs. For both classes, we study some of the basic properties of the Sturm‐Liouville theory. In the class of r‐CFSLPs, we discuss two types of CFSLPs which include left‐ and right‐sided CFDs, each of order α∈(n,n+1], and prove properties of the eigenvalues and the eigenfunctions in a certain Hilbert space. Also, we apply a fixed‐point theorem for proving the existence and uniqueness of the eigenfunctions. As an operator for the class of s‐CFSLPs, we first derive two fractional types of the hypergeometric differential equations of order α∈(0,1] and obtain their analytical eigensolutions as Gauss hypergeometric functions. Afterwards, we define the conformable fractional Legendre polynomial/functions (CFLP/Fs) as Jacobi polynomial and investigate their basic properties. Moreover, the conformable fractional integral Legendre transforms (CFILTs) based on CFLP/Fs‐I and ‐II are introduced, and using these new transforms, an effective procedure for solving explicitly certain ordinary and partial conformable fractional differential equations (CFDEs) are given. Finally, as a theoretical application, some fractional diffusion equations are solved.     

On calculus of local fractional derivatives

Local fractional derivative (LFD) operators have been introduced in the recent literature (Chaos 6 (1996) 505-513). Being local in nature these derivatives have proven useful in studying fractional differentiability properties of highly irregular and nowhere differentiable functions. In the present paper we prove Leibniz rule, chain rule for LFD operators. Generalization of directional LFD and multivariable fractional Taylor series to higher orders have been presented.     

Weber and Beltrami integrals of squared spherical Bessel functions: finite series evaluation and high‐index asymptotics

Weber integrals and Beltrami integrals are studied, which arise in the multipole expansions of spherical random fields. These integrals define spectral averages of squared spherical Bessel functions with Gaussian or exponentially cut power‐law densities. Finite series representations of the integrals are derived for integer power‐law index μ, which admit high‐precision evaluation at low and moderate Bessel index n. At high n, numerically tractable uniform asymptotic approximations are obtained on the basis of the Debye expansion of modified spherical Bessel functions in the case of Weber integrals. The high‐n approximation of Beltrami integrals can be reduced to Legendre asymptotics. The Airy approximation of Weber and Beltrami integrals is derived as well, and numerical tests are performed over a wide range of Bessel indices by comparing the exact finite series expansions of the integrals with their high‐index asymptotics. Copyright © 2013 John Wiley & Sons, Ltd.     

Fractional differential and integral operations via cumulative approach

In this study, a fractal operator model of cumulative processes is described. Accordingly, differential and integral operators of the fractional calculus are derived by the fractal operator model of a cumulative process. In order to exhibit the relation between our cumulative approach and fractional calculus, vertical motion of a body is handled within these frameworks. Thereby, regard to our assessments, the underlying physical mechanism of the success of the fractional differintegral operators in describing stochastic complex systems is uncovered to some extent.     

A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus

In this paper, we develop a fractional integro‐differential operator calculus for Clifford algebra‐valued functions. To do that, we introduce fractional analogues of the Teodorescu and Cauchy‐Bitsadze operators, and we investigate some of their mapping properties. As a main result, we prove a fractional Borel‐Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge‐type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann‐Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.     

Properties of the solutions of the fourth-order Bessel-type differential equation

The structured Bessel-type functions of arbitrary even-order were introduced by Everitt and Markett in 1994; these functions satisfy linear ordinary differential equations of the same even-order. The differential equations have analytic coefficients and are defined on the whole complex plane with a regular singularity at the origin and an irregular singularity at the point of infinity. They are all natural extensions of the classical second-order Bessel differential equation. Further these differential equations have real-valued coefficients on the positive real half-line of the plane, and can be written in Lagrange symmetric (formally self-adjoint) form. In the fourth-order case, the Lagrange symmetric differential expression generates self-adjoint unbounded operators in certain Hilbert function spaces. These results are recorded in many of the papers here given as references. It is shown in the original paper of 1994 that in this fourth-order case one solution exists which can be represented in terms of the classical Bessel functions of order 0 and 1. The existence of this solution, further aided by computer programs in Maple, led to the existence of a linearly independent basis of solutions of the differential equation. In this paper a new proof of the existence of this solution base is given, on using the advanced theory of special functions in the complex plane. The methods lead to the development of analytical properties of these solutions, in particular the series expansions of all solutions at the regular singularity at the origin of the complex plane.     

The Fourth-order Bessel–type Differential Equation

The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane.

There is a Bessel-type differential equation for each even-order integer; the equation of order two is the classical Bessel differential equation. These even-order Bessel-type equations are not formal powers of the classical Bessel equation.

When the independent variable of these equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coefficients. Embedded in this form of the equation is a spectral parameter; this combination leads to the generation of self-adjoint operators in a weighted Hilbert function space. In the second-order case one of these associated operators has an eigenfunction expansion that leads to the Hankel integral transform.

This article is devoted to a study of the spectral theory of the Bessel-type differential equation of order four; considered on the positive real axis this equation has singularities at both end-points. In the associated Hilbert function space these singular end-points are classified, the minimal and maximal operators are defined and all associated self-adjoint operators are determined, including the Friedrichs self-adjoint operator. The spectral properties of these self-adjoint operators are given in explicit form.

From the properties of the domain of the maximal operator, in the associated Hilbert function space, it is possible to obtain a virial theorem for the fourth-order Bessel-type differential equation.

There are two solutions of this fourth-order equation that can be expressed in terms of classical Bessel functions of order zero and order one. However it appears that additional, independent solutions essentially involve new special functions not yet defined. The spectral properties of the self-adjoint operators suggest that there is an eigenfunction expansion similar to the Hankel transform, but details await a further study of the solutions of the differential equation.     

Numerical solution of the Bagley–Torvik equation by the Bessel collocation method

In this article, a numerical technique is presented for the approximate solution of the Bagley–Torvik equation, which is a class of fractional differential equations. The basic idea of this method is to obtain the approximate solution in a generalized form of the Bessel functions of the first kind. For this purpose, by using the collocation points, the matrix operations and a generalization of the Bessel functions of the first kind, this technique transforms the Bagley–Torvik equation into a system of the linear algebraic equations. Hence, by solving this system, the unknown Bessel coefficients are computed. The reliability and efficiency of the proposed scheme are demonstrated by some numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.     

On wave diffraction by a half‐plane with different face impedances

The impedance wave diffraction problem by a half‐plane screen is revisited in view of its well‐posedness upon different impedance and wave parameters. The problem is analysed with the help of potential and pseudo‐differential operators. Seven conditions between the impedance and wave numbers are found under which the problem will be well‐posed in Bessel potential spaces. In addition, an improvement of the regularity of the solutions is shown for the previous seven conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

New formulae for higher order derivatives and applications

We present new formulae (the Slevinsky–Safouhi formulae I and II) for the analytical development of higher order derivatives. These formulae, which are analytic and exact, represent the kth derivative as a discrete sum of only k+1 terms. Involved in the expression for the kth derivative are coefficients of the terms in the summation. These coefficients can be computed recursively and they are not subject to any computational instability. As examples of applications, we develop higher order derivatives of Legendre functions, Chebyshev polynomials of the first kind, Hermite functions and Bessel functions. We also show the general classes of functions to which our new formula is applicable and show how our formula can be applied to certain classes of differential equations. We also presented an application of the formulae of higher order derivatives combined with extrapolation methods in the numerical integration of spherical Bessel integral functions.     

On evolutionary equations with material laws containing fractional integrals

A well‐posedness result for a time‐shift invariant class of evolutionary operator equations involving material laws with fractional time‐integrals of order α ? ]0, 1[ is considered. The fractional derivatives are defined via a function calculus for the (time‐)derivative established as a normal operator in a suitable L² type space. Employing causality, we show that the fractional derivatives thus obtained coincide with the Riemann‐Liouville fractional derivative. We exemplify our results by applications to a fractional Fokker‐Planck equation, equations describing super‐diffusion and sub‐diffusion processes, and a Kelvin‐Voigt type model in fractional visco‐elasticity. Moreover, we elaborate a suitable perspective to deal with initial boundary value problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Groups of partial differential operators and the generalized Bessel functions

Considering the generalized Bessel functions satisfying a special ordinary differential equation of mth order, we derive some addition theorems and generating functions with the help of some algebra constructed for a group of first order partial differential operators grounding on the recurrence relations for these functions.     

On a Certain Class of Integral Equations Associated with Hankel Transforms

This paper deals with a new class of Fredholm integral equations of the first kind associated with Hankel transforms of integer order. Analysis of the equations is based on operators transforming Bessel functions of the first kind into kernels of Weber–Orr integral transforms. Their inverse operators are established by means of new inversion theorems for the Hankel and Weber–Orr integral transforms of functions belonging to L ¹ and L ². These operators together with the proven Paley–Wiener’s theorem for the Weber–Orr transform enable to regularize the equations and, in special cases, to derive explicit solutions. The integral equations analyzed in this paper can be employed instead of dual integral equations usually treated with the Cooke–Lebedev method. An example manifests that it may be preferable because of the possibility to control norms of operators in the regularized equations.      

Existence of fractional differential chains and factorizations based on transformations

In this work, we deal with the existence of the fractional integrable equations involving two generalized symmetries compatible with nonlinear systems. The method used is based on the Bä cklund transformation or B‐transformation. Furthermore, we shall factorize the fractional heat operator in order to yield the fractional Riccati equation. This is done by utilizing matrix transform Miura type and matrix operators, that is, matrices whose entries are differential operators of fractional order. The fractional differential operator is taken in the sense of Riemann–Liouville calculus. Copyright © 2014 John Wiley & Sons, Ltd.     

Application of pseudo‐trigonometry to Bessel series and integrals of Bessel functions

In this article, an extension of the Laplace transform of J_n (t) to pseudo‐trigonometric function is discussed. We are seeking elementary functions expressed by Bessel series. It is shown that the result is applicable to the solution of the first‐order differential equation. The expression of modified Bessel integral formulas in pseudo‐trigonometric function is also discussed.     

Eigenvalue problems for a non‐self‐adjoint Bessel‐type operators in limit‐point case

It is shown in the Weyl limit‐point case that system of root functions of the non‐self‐adjoint Bessel operator and its perturbation Sturm–Liouville operator form a complete system in the Hilbert space. Furthermore, asymptotic behavior of the eigenvalues of the non‐self‐adjoint Bessel operators is investigated, and it is proved that system of root functions form a Bari basis in the same Hilbert space. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical solution of nonlinear partial quadratic integro‐differential equations of fractional order via hybrid of block‐pulse and parabolic functions

In this paper, an effective numerical approach based on a new two‐dimensional hybrid of parabolic and block‐pulse functions (2D‐PBPFs) is presented for solving nonlinear partial quadratic integro‐differential equations of fractional order. Our approach is based on 2D‐PBPFs operational matrix method together with the fractional integral operator, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved, and the solution of fractional nonlinear partial quadratic integro‐differential equations is achieved. Convergence analysis and an error estimate associated with the proposed method is obtained, and it is proved that the numerical convergence order of the suggested numerical method is O(h³) . The validity and applicability of the method are demonstrated by solving three numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the exact solutions much easier.     

Non‐self‐adjoint Bessel and Sturm–Liouville boundary value problems in limit‐circle case

It is shown in the limit‐circle case that system of root functions of the non‐self‐adjoint maximal dissipative (accumulative) Bessel operator and its perturbation Sturm–Liouville operator form a complete system in the Hilbert space. Furthermore, asymptotic behavior of the eigenvalues of the maximal dissipative (accumulative) Bessel operators is investigated, and it is proved that system of root functions form a basis (Riesz and Bari bases) in the same Hilbert space. Copyright © 2014 John Wiley & Sons, Ltd.     

Index transforms with the squares of Bessel functions

New index transforms, involving the squares of Bessel functions of the first kind as the kernel, are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution to the initial value problem for the third-order partial differential equation, involving the Laplacian, is obtained.     

The discrete duality finite volume method for a class of quasi‐Newtonian Stokes flows

In this paper, we propose a discrete duality finite volume (DDFV) scheme for the incompressible quasi‐Newtonian Stokes equation. The DDFV method is based on the use of discrete differential operators which satisfy some duality properties analogous to their continuous counterparts in a discrete sense. The DDFV method has a great ability to handle general geometries and meshes. In addition, every component of the velocity gradient can be reconstructed directly, which makes it suitable to deal with the nonlinear terms in the quasi‐Newtonian Stokes equation. We prove that the proposed DDFV scheme is uniquely solvable and of first‐order convergence in the discrete L²‐norms for the velocity, the strain rate tensor, and the pressure, respectively. Ample numerical tests are provided to highlight the performance of the proposed DDFV scheme and to validate the theoretical error analysis, in particular on locally refined nonconforming and polygonal meshes.