Author : Muhammad Nadeem, Afifa Maqsood, Muhammad Moaz Safer, Hira Fatima, Abdul Raheem and Tazeem Fatima

Comparison of Numerical Techniques for the solutions of Nonlinear Singular Differential Equations is a very
important topic in applied mathematics. In this work, we present a powerful method for the numerical solution of non-linear singular initial value problems. We calculate approximate solutions for nonlinear singular initial value problems by using modified Laplace decomposition method and also by the use of modified adomian decomposition method. In this work, we present a powerful method for the numerical solution of non-linear singular boundary value problems, namely the advanced Adomian decomposition method which is the modification of adomian decomposition method. In this method, we use all the boundary conditions to obtain the coefficient of the approximate series solution. Moreover, convergence analysis and an error bound for the approximate solution are discussed. This method overcomes the singular behaviour of problems and illustrates the approximations of high precision with a large effective region of convergence. To prove the robustness and effectiveness of the proposed method, various examples are considered and the obtained results are compared with the other existing methods. This method overcomes the singular behaviour of problems and illustrates the approximations of high precision with a large effective region of convergence. To prove the effectiveness of the proposed method, various examples are considered and the obtained results are compared with the other existing methods. For the accuracy of results we have calculated the solution up to more than three places.it overcome the singular behaviour of problems and also find the approximation with high accuracy by using these methods. We will find the effectiveness or pact of proposed methods. Different example is taken and the obtained results are compared with the previous researched. Error estimation and convergence order for the presented method are also discussed. Our numerical experiments validate our theoretical findings and demonstrate the performance of the algorithm in terms of simplicity, accuracy, and efficiency.

Keywords: Numerical Techniques, Adomian Decomposition Method, Differential Transformation Method