IN THIS PAPER, WE USE THE BERNOULLI Operational matrix OF DERIVATIVES AND THE COLLOCATION POINTS, FOR SOLVING LINEAR AND NONLINEAR OPTIMAL CONTROL PROBLEMS (OCPS). BY BERNOULLI POLYNOMIALS BASES, THE TWO-POINT BOUNDARY VALUE PROBLEM (TPBVP), DERIVED FROM THE PONTRYAGINS MAXIMUM PRINCIPLE, TRANSFORMS INTO THE matrix EQUATION.

IN THIS PAPER, WE PRESENT A NEW TECHNIQUE FOR SOLVING NUMERICALLY LANGEVIN EQUATION BASED ON Operational matrix AND STOCHASTIC Operational matrix. NUMERICAL SIMULATIONS ARE PRESENTED TO ILLUSTRATE OUR MATHEMATICAL FINDINGS.

AN EFFICIENT NUMERICAL METHOD IS PROPOSED FOR SOLVING NONLINEAR STOCHASTIC DIFFERENTIAL EQUATIONS, USING Operational matrix OF BLOCK PULSE FUNCTIONS. BY USING THIS APPROACH, THE STOCHASTIC DIFFERENTIAL EQUATION REDUCES TO A NONLINEAR SYSTEM OF ALGEBRAIC EQUATIONS WHICH CAN BE SOLVED BY NEWTON, S ITERATIVE METHOD. ACCURACY AND EFFICIENCY OF THE METHOD ARE SHOWN WITH AN EXAMPLE.

IN THIS PAPER, WE FIRST INTRODUCE BLOCK PULSE FUNCTIONS AND THE BLOCK PULSE Operational MATRICES OF THE FRACTIONAL ORDER INTEGRATION. ALSO THE BLOCK PULSE Operational MATRICES OF THE FRACTIONAL ORDER DIFFERENTIATION ARE OBTAINED.THEN WE PRESENT A COMPUTATIONAL METHOD BASED ON THE ABOVE RESULTS FOR SOLVING A CLASS OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS.

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.