Abstract
In this paper, we consider two different systems of triple q-integral equations, where the kernel is a third Jackson q-Bessel function. The solution of the first system is reduced to two simultaneous Fredholm q-integral equations of the second kind. We solve the second system by using the solution of specific dual q-integral equations. Some examples are included.
Highlights
In the past years, several authors have described various methods to solve triple integral equations especially of the form ∞A(u)K(u, x) du = f (x), < x < a, w(u)A(u)K(u, x) du = g(x), a < x < b, A(u)K(u, x) du = h(x), b < x < ∞, where w(u) is the weight function, K(u, x) is the kernel function
Some mixed boundary value problems of the mathematical theory of elasticity are solved by reducing them to multiple integral equations
Harmonic shear oscillations of a rigid stamp with a plane base coupled to an elastic half-space were studied in [ ] and reduced to dual integral equations
Summary
Several authors have described various methods to solve triple integral equations especially of the form. We use a result from [ ] for a solution of dual q -integral equations to solve triple q -integral equations. We end this section by introducing some q-fractional operators that we use in solving the triple q-integral equations under consideration. The technique of using fractional operators in solving dual and triple integral equations is not new. In [ ], the authors introduced a slight modification of the operator Kqη,α This operator is denoted by Kqη,α and defined by (x/t; q)α– t–η– φ(qt) dqt, x where α = – , – , . Note that this operator satisfies the following semigroup identity: KqαKqβ φ(x) = Kqα+β φ(x) for all α and β.
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