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In this paper, Reduced Differential Transform method in the framework of (p, q)-calculus, denoted by Rp,qDT , has been introduced and applied in solving a variety of differential equations such as diffusion equation, 2Dwave equation, K-dV equation, Burgers equations and Ito system. While the diffusion equation has been studied for the special case p = 1, i.e., in the framework of q-calculus, the other equations have not been studied even in q-calculus.

In this paper, we introduce and study a (p, q)-Mellin transform and its corresponding convolution and inversion. In terms of applications of the (p, q)-Mellin transform, we solve some integral equations. Moreover, a (p, q)-analogue of the Titchmarsh theorem is also derived.

In this paper, we have introduced and studied finite $$(p,q)$$ -Mellin transform which extends the known theory of finite q-Mellin transform. Also, in this paper, we study finite $$(p,q)$$ -Mellin transform in the framework of symmetric calculus. This study is new even for the special case $$p=q$$ . In both the cases, we provide several basic properties, inversion formulas, the appropriate convolution product and the Parseval type relations.

In this paper, we have introduced and studied two-dimensional [Formula: see text]-Mellin transform which extends the known results of two-dimensional [Formula: see text]-Mellin transform. We provide its several basic properties, the appropriate convolution, the inversion formula and the Parseval-type relations. Some applications of [Formula: see text]-Mellin transform have been pointed out in solving integral equations. Finally, a Titchmarsh-type theorem has been proved.

In this paper a new generalization of $B^*$-continuity has been defined, to be called as $B^*$-clopen continuity and many of its properties have been studied including the convergence behaviour of sequences of such functions. Also a new type of oscillation function, called cl-oscillation has been defined, which characterizes the $B^*$-clopen continuity.