AN EFFECTIVE METHOD BASED ON Triangular functions IS PROPOSED FOR THE SOLUTIONS OF 2D VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS. THE MAIN ADVANTAGE OF THIS METHOD IS ITS EFFICIENCY AND SIMPLE APPLICABILITY. A NUMERICAL EXAMPLE IS PRESENTED TO SHOW THAT THE ACCURACY OF THE METHOD.

A REAL MATRIX R WITH NONNEGATIVE ENTRIES IS CALLED ROW SUB STOCHASTIC IF EACH ROW SUM ISAT MOST 1. FOR X, Y Î RN, WE SAY THAT X IS ROW SUB STOCHASTIC UPPER Triangular MAJORIZED BY Y (WRITE AS X≺SUT Y) IF THERE EXISTS A ROW SUB STOCHASTIC UPPER Triangular MATRIX R SUCH THAT X=RY. IN THIS PAPER, THE STRUCTURE OF ALL LINEAR functions T: R3 → R3, PRESERVING (STRONGLY PRESERVING)≺SUT ARE CHARACTERIZED.

IN THIS PAPER, WE STATE AND PROVE A NEW FORMULA EXPRESSING EXPLICITLY THE RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL OF THE Triangular functions (TFS) WITH ANY FRACTIONAL-ORDER, IN TERMS OF TFS THEMSELVES AND USE IT TO CONSTRUCT A NEW AND GENERAL FORMULATION FOR THE Triangular FUNCTION (TF) OPERATIONAL MATRIX OF FRACTIONAL INTEGRAL (TF-OMFI). ALSO, BY USING THE TF-OMFI AND A SPECTRAL TAU METHOD, WE DEVELOP A DIRECT SOLUTION TECHNIQUE FOR SOLVING A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION.

IN THIS PAPER WE INTEND TO OFFER A NUMERICAL METHOD TO SOLVE LINEAR FUZZY FREDHOLM INTEGRAL EQUATIONS SYSTEM OF THE SECOND KIND. THIS METHOD CONVERTS THE GIVEN FUZZY SYSTEM INTO A LINEAR SYSTEM OF ALGEBRAIC EQUATIONS BY USING Triangular ORTHOGONAL functions. THE PROPOSED METHOD IS ILLUSTRATED BY AN EXAMPLE AND ALSO RESULTS ARE COMPARED WITH THE EXACT SOLUTION BY USING COMPUTER SIMULATIONS.

IN THIS PAPER WE INTEND TO OFFER A NUMERICAL METHOD TO SOLVE LINEAR TWO-DIMENSIONAL FREDHOLM INTEGRAL EQUATIONS SYSTEM OF THE SECOND KIND. THIS METHOD CONVERTS THE GIVEN TWO-DIMENSIONAL FREDHOLM INTEGRAL EQUATIONS SYSTEM INTO A LINEAR SYSTEM OF ALGEBRAIC EQUATIONS BY USING TWO DIMENSIONAL Triangular functions. MOREOVER, WE PROVE THE CONVERGENCE OF THE METHOD. FINALLY THE PROPOSED METHOD IS ILLUSTRATED BY AN EXAMPLE AND ALSO RESULTS ARE COMPARED WITH THE EXACT SOLUTION BY USING COMPUTER SIMULATIONS.

THIS PAPER GIVES EFFICIENT METHOD FOR SOLVING FINANCIAL PROBLEMS FORMULATED BY STOCHASTIC VOLTERRA INTEGRAL EQUATIONS. HERE, WE CONSIDER Triangular functions AND THEIR OPERATIONAL MATRIX AND STOCHASTIC OPERATIONAL MATRIX OF INTEGRATION. THIS METHOD HAS SEVERAL ADVANTAGES IN REDUCING COMPUTATIONAL BURDEN WITH GOOD DEGREE OF ACCURACY.