Solution of the tumor-immune system by differential transform method

Abstract

In this paper, differential transform method (DTM) is presented to solve Tumor-immune system at two initial conditions where two different cases of the interaction between tumor cells and effector cells. The system is presented to show the ability of the method for non-linear systems of differential equations. By using small iteration, the results of DTM are near the results of Runge-Kutta fourth-fifth order method (ode45 solver in MATLAB) and better than the results of Runge-Kutta second-third order method (ode23 solver in MATLAB). Also, the residual error of DTM's solutions approach zero. Therefore, DTM's solutions approximate exact solutions. Finally, we conclude formulae that we can find DTM's solutions, better than the results of Runge-Kutta second-third order method, in any interval we need.

Dynamic Response Analysis of Structures Using Legendre–Galerkin Matrix Method

“…Since 1994, the Chebyshev polynomials [7], Legendre polynomials [8], Bessel polynomials [9], Hermite polynomials [10], Laguerre polynomials [11], and matrix methods have been used in many research studies to solve linear and nonlinear equations with high orders including partial differential equations, hyperbolic partial differential equations, delay equations, integral and integro-differential equations, SDOF and MDOF systems, etc. One of the approaches to find the solution to an initial value problem is taking semi-analytical procedures such as the differential transform method (DTM) [12][13][14][15][16][17][18][19][20][21][22][23]. The nature of dynamic equations of motion of SDOF systems makes them differential equations with initial values; hence, DTM has been also applied to solve non-linear SDOF problems [24][25][26][27][28][29][30][31][32][33].…”