Abstract
A method is developed for solutions of two sets of triple integral equations involving associated Legendre functions of imaginary arguments. The solution of each set of triple integral equations involving associated Legendre functions is reduced to a Fredholm integral equation of the second kind which can be solved numerically.
1. Introduction
Dual integral equations involving Legendre functions have been solved by Babloian [1]. He applied these equations to problems of potential theory and to a torsion problem. Later on Pathak [2] and Mandal [3] who considered dual integral equations involving generalized Legendre functions which have more general solution than the ones considered by Babloian [1]. Recently, Singh et al. [4] considered dual integral equations involving generalized Legendre functions, and their results are more general than those in [1–3].
In the analysis of mixed boundary value problems, we often encounter triple integral equations. Triple integral equations involving Legendre functions have been studied by Srivastava [5]. Triple integral equations involving Bessel functions have also been considered by Cooke [6–9], Tranter [10], Love and Clements [11], Srivastava [12], and most of these authors reduced the solution into a solution of Fredholm integral equation of the second kind. The relevant references for dual and triple integral equations are given in the book of Sneddon [13].
In this paper, a method is developed for solutions of two sets of triple integral equations involving generalized Legendre functions in Sections 3 and 4. Each set of triple integral equations is reduced to a Fredholm integral equation of the second kind which may be solved numerically. The aim of this paper is to find a more general solution for the type of integral equations given in [1–5] and to develop an easier method for solving triple integral equations in general.
2. Integral Involved Generalized Legendre Functions and Some Useful Results
We first summarize some known results needed in the paper.
We find from [14, equation (21), page 330] thatwhere and from [4], we obtainwhere and denotes the Heaviside unit function. Furthermore, , and is the generalized Legendre function defined in [15, page 370]. From [4, 16], the generalized Mehler-Fock transform is defined byand its inversion formula isEquations (2.1) and (2.2) are of form (2.3). From the inversion formula given by (2.4), (2.1), and (2.2), it follows that
The inversion theorem for Fourier cosine transforms and the results (2.1) and (2.2) lead to
If is monotonically increasing and differentiable for and in this interval, then the solutions of the equationsare given by Sneddon [13] asrespectively, where the prime denotes the derivative with respect to .
3. Triple Integral Equations with Generalized Legendre Functions: Set I
In this section, we will find solution of the following triple integral equations: where is an unknown function to be determined, is a known function, and is the generalized Legendre function defined in Section 2 and , , .
The trial solution of (3.1), (3.2), and (3.3) can be written aswhere is an unknown function to be determined. On integrating (3.4) by parts, we getwhere the prime denotes the derivative with respect to .
Substituting (3.5) into (3.3), interchanging the order of integrations and using (2.2), we find that (3.3) is satisfied identically. Substituting (3.5) into (3.1) and using the integral defined by (2.2), we obtainEquation (3.6) is equivalent to the following integral equation:By substituting (3.4) into (3.2), interchanging the order of integrations and using the integral defined by (2.1) we find thatFor obtaining the solution of the problem, we need to solve two Abel's type integral equations (3.7) and (3.8).
We assume thatThe above equation is of the same form as (3.7) and defined in a different region. Equation (3.9) is of form (2.12). Hence, the solution of the integral equation (3.9) can be written as
The solution of Abel's type integral equations (2.11) together with (3.7) and (3.9) leads to
Equations (3.10) and (3.11) mean that (3.7) is satisfied identically. Equation (3.8) can be rewritten in the formSubstituting the expression for from (3.11) and (3.10) into the first and second integral of (3.12) we obtainwhere
Assuming that the right-hand side of (3.13) is a known function of it has the form of (2.9), whose solution is given bywhereFrom the integralwe then obtainEquation (3.14) is an Abel-type equation. Hence, its solution isSubstituting the expression for from (3.20) into (3.21), integrating by parts, and finally interchanging the order of integrations in second integral, we arrive atThe integral together with (3.22) leads toFrom (3.19), (3.21), and (3.24), we obtainwhereFrom (3.25), (3.16) can be written asEquation (3.27) is a Fredholm integral equation of the second kind with kernel . The kernel is defined by (3.26). The integral in (3.26) cannot be solved analytically, but for particular values of and the values of can be found numerically. Hence, the numerical solution of Fredholm integral equation (3.27) can be obtained for particular value of , , and to find numerical values of . Making use of (3.20), (3.11), and (3.10), the numerical results for can be obtained. Finally, making use of (3.4) the numerical results for can be obtained.
4. Triple Integral Equations with Generalized Legendre Functions: Set II
In this section, we will find the solution of the following triple integral equations: where , , .
We assume thatThe inversion formula for generalized Mehler-Fock transforms (2.4) together with (4.3) and (4.4) implies that
Multiplying (4.1) by , integrating both sides from 0 to and with respect to , and then using (2.6) we obtainSubstituting the value of from (4.5) into (4.6), interchanging the order of integrations, and using the integral (2.2), we getSubstituting the value of from (4.5) into (4.2) and interchanging the order of integrations we arrive atwhereand then (2.8) and (2.2) imply that
Equation (4.7) is an Abel-type equation and has the form (2.9). Hence, the solution of (4.7) is
Using (4.10) and (2.5), (4.8) can be written in the formUsing the formulawe can write (4.12) in the formwhere
Assuming that the right-hand side of (4.14) is known function equation and (4.14) has the form of (2.10), hence the solution of (4.14) can be written aswhere
Equation (4.17) is simplified toLetEquation (4.15) is of the form of (2.9). Hence, its solution is
Substituting the expression for from (4.20) into (4.19) and integrating by parts and then using the following integral:we find thatMaking use of (4.18), (4.19), and (4.22), we find thatwhereUsing (4.17) and (4.23), (4.16) can be written in the form
Equation (4.25) is a Fredholm integral equation of the second kind with kernel defined by (4.24). The Fredholm integral equation (4.25) may be solved to find numerical values of for particular values of . And hence from (4.20) and (4.5), the numerical values for can be obtained for particular values of , , and .
5. Conclusions
The solution of the two sets of triple integral equations involving generalized Legendre functions is reduced to the solution of Fredholm integral equations of the second kind which can be solved numerically.