A numerical method for solving two-dimensional nonlinear Volterra fuzzy integral equation (2D-NVFIE) of the second kind will is introduced. We convert a nonlinear Volterra fuzzy integral equation to a nonlinear system of Volterra integral equation in crisp case. We use Adomian Decomposition Method (ADM) to find the approximate solution of this system and hence obtain an approximation for fuzzy solution of the nonlinear Volterra fuzzy integral equation. Also, the existence and uniqueness of the solution and convergence of the proposed method are proved. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique.

In this paper, we combine double Sumudu transform with Adomian decomposition method to solve two dimensional fuzzy convolution Volterra integral equations. We give some definition for fuzzy number, fuzzy valued function which are prerequest for combine the double fuzzy Sumudu transform with Adomian decomposition transform method. We describe the method and finally, numerical examples are given to illustrate the efficiency and applicability of our method.

In this paper we present some properties of double \(q\)-Sumudu transform in \(q\)-calculus by using the functions of two variables. Furthermore results on convergence, absolute convergence and convolution are discussed. At the end some examples are given to illustrate use of double \(q\)-Sumudu transform.

We introduce finite-observation conditional guarded language operators on the complete length-based ultrametric space of formal languages over a finite alphabet. A finite observation map selects a guarded branch according to the membership of prescribed test words. Although each branch is Banach-contractive, branch switching may destroy global Banach contractivity. We derive depth–residual conditions yielding a max-type Kannan inequality with an explicit constant $\alpha <1/2$. Consequently, the operator has a unique fixed language, and the Picard iteration converges from every initial language. We also construct multi-branch operators that are Kannan-contractive but not Banach contractive and obtain residual error estimates, explicit convergence bounds, finite-depth certification, and eventual stabilization of the selected branch.