Coupled of Semi Analytic Approach Associated with Laplace Transform First Step for Solving Matrix Differential Equations Quadratic Form when the Time-Delay in Noise Term (Baghdad Science Journal, 2024)
A novel technique and an efficient modification based on Adomian decomposition method and homotopy approach for finding accurately analytic solutions to non-linear (noise term) quadratic matrix retarded delay equations connected with the method of steps to make the problem more easily is discussed. These approaches more efficiently, effectively and accurately. Wholly integration for homotopy analysis method use in state the wholly integration for Adomian approach. Main advantage of this technique is to get more an accurate and efficient results with more extended of the convergence region of iterative approximate solutions obtained with bigger and whole time interval and to know the accurate solution with long interval under delay influence until    and can more. Term of delay is disappeared after apply the method of steps. Absolute residual error is conducted. To reduce the time and more complicated calculations, Laplace transform for each components is applied. Finally, the results which obtained by this technique is an effective and rapidly converge for exact solution for whole time interval with more extended of the convergence region. This technique can used to different nonlinear problem. The Adomian decomposition method is a semi analytical technique for solving different type of differential equations ordinary, partial, fractional, delay differential equations and many type. This method was developed by George Adomian. It is rapidly converge to exact solution and used for linear , nonlinear, homogeneous and nonhomogeneous equations. Adomian polynomial allow the solution converge to exact solution without simply linearizing the problem under consideration. The same for homotopy.