On some integral equations with the generalized Legendre function

We solve and investigate an integral equation with the generalized associated Legendre functionP _k ^m,n (z) by using the fractional integro-differential calculus. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 263–267, February, 1999.     

On an integral transform with generalized associated Legendre functions

We study some new properties of generalized associated Legendre functions of first and second kind P _k ^m,n (z) and Q _k ^m,n (z). Applying these functions, we introduce an integral transform that can be used in solving boundary-value problems of mathematical physics.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 33–43.     

On a class of dual integral equations involving trigonometrical functions

In this paper we consider certain dual integral equations involving trigonometrical functions whose closed form solutions are obtained. Solutions are obtained by using the properties of Mehler-Fock transforms and the inversion theorem of the generalized Mehler-Fock transforms. The solutions of these dual integral equations have applications in engineering.     

On dual and triple integral equations involving modified bessel functions

A generalisation of some tripe integral equations occuring in the solution of certain mixed boundary value problems involving the wave equation is investigated. It is found that the solution of the equations can be expressed in terms of the solution of a Fredholm intgral equation of the second kind and that a related pair of dual integral equations can be solved exactly     

Solution of one class of systems of dual summation equations for associated Legendre functions

We have obtained analytical solutions of one class of systems of dual summation equations for associated Legendre functions with fractional indices. Such equations appear in studying the interaction of vector electromagnetic fields with the circular edge of a conductive open cone in the low-frequency region. We have derived formulas for the reexpansion of Legendre functions, which are used for passage from summation equations to infinite systems of linear algebraic equations, containing convolution-type matrix operators. The operators inverse to them are applied for finding a solution in the required class of sequences. We give an example of determining the effect of interaction of TM- and TE-waves with the edge of a finite cone.     

A Numerical solution to the linear and nonlinear Fredholm integral equations using Legendre wavelet functions

Two methods for solving the Fredholm integral equation of the second kind in linear case, i.e. f (x) – λ ∫_a^b K (x,y)f (y)dy = g (x), and nonlinear case, i.e., f (x) = g (x) + λ ∫_a^b K (x,y)F (f (y))dy, are proposed. In order to solve the linear equation, the kernel K (x,y) as well as the functions f and g are initially approximated through Legendre wavelet functions. This leads to a system of linear equations its solution culminates in a solution to the Fredholm integral equation. In nonlinear case only K (x,y) is approximated by Legendre wavelet base functions. This leads to a separable kernel and makes it possible to employ a number of earlier methods in solving nonlinear Fredholm integral equation with separable kernels. Another feature of the proposed method is that it finds the solution as a function instead of specific solution points, what is done by the majority of the existing methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Legendre spectral Galerkin method for second-kind Volterra integral equations

The Legendre spectral Galerkin method for the Volterra integral equations of the second kind is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L ₂ norm) will decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical examples are given to illustrate the theoretical results.      

Numerical solutions of integral and integro-differential equations using Legendre polynomials

In this paper, a finite Legendre expansion is developed to solve singularly perturbed integral equations, first order integro-differential equations of Volterra type arising in fluid dynamics and Volterra delay integro-differential equations. The error analysis is derived. Numerical results and comparisons with other methods in literature are considered.      

Numerical solution of nonlinear integral equations using alternative Legendre polynomials

In this paper, the operational matrices of integration and the product for the alternative Legendre polynomials (ALPs) are first derived. Then, using these operational matrices and the collocation method, the nonlinear Volterra–Fredholm–Hammerstein integral equations are reduced to a set of nonlinear algebraic equations with unknown ALP coefficients. Some error estimations are provided and the efficiency and accuracy is verified by applying the method to some examples chosen from other literature.     

Fractional-order Legendre functions for solving fractional-order differential equations

In this article, a general formulation for the fractional-order Legendre functions (FLFs) is constructed to obtain the solution of the fractional-order differential equations. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, an efficient and reliable technique for the solution of them is too important. For the concept of fractional derivative we will adopt Caputo’s definition by using Riemann–Liouville fractional integral operator. Our main aim is to generalize the new orthogonal functions based on Legendre polynomials to the fractional calculus. Also a general formulation for FLFs fractional derivatives and product operational matrices is driven. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

On Legendre functions of imaginary degree and associated integral transforms

New integral representations, asymptotic formulas, and series expansions in powers of tanh(t/2) are obtained for the imaginary and real parts of the Legendre function P_iξ(cosht). Coefficients of these series expansions are orthogonal polynomials in the real variable ξ. A number of relations for these orthogonal polynomials are obtained on the basis of the generating function. Several inversion theorems are proven for the integral transforms involving the Legendre function of imaginary degree. In many cases it is preferable to employ these transforms, than Mehler-Fok transforms, since conditions placed on functions are less restrictive.     

On the solution of dual integral equations

A quick method of solution of dual integral equations involving a kernel comprised of trigonometric functions is explained. Certain solvability criteria are obtained in terms of forcing functions for the unique solution of the dual integral equations.     

On a class of dual integral equations

Dual integral equations involving generalized Legendre functions and certain trigonometrical functions are considered. Solutions are obtained by using properties of generalized Legendre functions and the inversion theorem for the generalized Mehler — Fock transform.     

Some properties of generalized associated Legendre functions of second kind

We investigate some new properties of generalized associated Legendre polynomials of the second kind, establish new relationships between these polynomials, construct differential operators with the functions P _k ^m,n (z), Q _k ^m,n (z), and describe some applications.Kiev Polytechnical Institute. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 34–45, 1989.