An ordinary differential equation approach for nonlinear programming and nonlinear complementary problem

We consider an ordinary differential equation (ODE) approach for solving non- linear programming (NLP) and nonlinear complementary problem (NCP). The Karush- Kuhn Tucker (KKT) optimality conditions can be converted to NCP. Based on the Fischer-Burmeister (FB) function and the Natural-Residual (NR) function are obtained the new NCP-functions. A special technique is employed to reformulate of the NCP as the system of nonlinear algebraic equations (NAEs) later on reformulated once more by force. of an original time-like function into an ODE. Afterwards, a group preserving scheme (GPS) is a package to reformulate an ODE into the new numerical equation in a way the ODEs system is designed into a nonlinear dynamical system (NDS) and is continued to a discovery the new numerical equation through activating the Lorentz group SO0(n, 1) and its Lie algebra so(n, 1). Lastly, the fictitious time integration method (FTIM) is utilized into this new numerical equation to determine an approximation solution at the numerical experiments area.

On Numerical Solution Of The Time Fractional Advection-Diffusion Equation Involving Atangana-Baleanu-Caputo Derivative

A powerful algorithm is proposed to get the solutions of the time fractional Advection-Diffusion equation(TFADE): $^{ABC}\mathcal{D}_{0^+,t}^{\beta}u(x,t) =\zeta u_{xx}(x,t)- \kappa u_x(x,t)+$ F(x, t), 0 < β ≤ 1. The time-fractional derivative $^{ABC}\mathcal{D}_{0^+,t}^{\beta}u(x,t)$is described in the Atangana-Baleanu Caputo concept. The basis of our approach is transforming the original equation into a new equation by imposing a transformation involving a fictitious coordinate. Then, a geometric scheme namely the group preserving scheme (GPS) is implemented to solve the new equation by taking an initial guess. Moreover, in order to present the power of the presented approach some examples are solved, successfully.

Ficitious time integration method for solving the time fractional gas dynamics equation

In this work a poweful approach is presented to solve the time-fractional gas dynamics equation. In fact, we use a fictitious time variable y to convert the dependent variable w(x, t) into a new one with one more dimension. Then by taking a initial guess and implementing the group preserving scheme we solve the problem. Finally four examples are solved to illustrate the power of the offered method.