The numerical solution of the second kind of Abel equations by the modified matrix-exponential method

In this paper, the modified matrix exponential method (MME), under the zero-order hold (ZOH) assumption, is applied to solve the Abel equation of the second kind. The modified exponential matrix method is iterative, and by increasing the iteration, we can get a better approximation with fewer errors. We use the MME to turn an Abel differential equation into a system of nonlinear equations and determine the solution. By using the MME, the Abel differential equations approximate well. Using the numerical results, we can conclude that this method is effective, and in comparison with well-known techniques, the MME is highly accurate.

Keywords:

Modified matrix exponential , Matrix exponential , Abel equation of the second kind , Nonlinear differential equations , Jacobian matrix