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The electrical impedance equation is considered an ill-posed problem where the solution to the forward problem is more easy to achieve than the inverse problem. This work tries to improve convergence in the forward problem method, where the Pseudoanalytic Function Theory by means of the Taylor series in formal powers is used, incorporating a regularization method to make a solution more stable and to obtain better convergence. In addition, we include a comparison between the designed algorithms that perform proposed method with and without a regularization process and the autoadjustment parameter for this regularization process.

This work is dedicated to the analysis of the forward and the inverse problem to obtain a better approximation to the Electrical Impedance Tomography equation. In this case, we employ for the forward problem the numerical method based on the Taylor series in formal power and for the inverse problem the Finite Element Method. For the analysis of the forward problem, we proposed a novel algorithm, which employs a regularization technique for the stability, additionally the parallel computing is used to obtain the solution faster; this modification permits to obtain an efficient solution of the forward problem. Then, the found solution is used in the inverse problem for the approximation employing the Finite Element Method. The algorithms employed in this work are developed in structural programming paradigm in C++, including parallel processing; the time run analysis is performed only in the forward problem because the Finite Element Method due to their high recursive does not accept parallelism. Some examples are performed for this analysis, in which several conductivity functions are employed for two different cases: for the analytical cases: the exponential and sinusoidal functions are used, and for the geometrical cases the circle at center and five disk structure are revised as conductivity functions. The Lebesgue measure is used as metric for error estimation in the forward problem, meanwhile, in the inverse problem PSNR, SSIM, MSE criteria are applied, to determine the convergence of both methods.

In order to protect the information, the study and development of a new cryptographic method is a hard duty, due to the high amount of techniques to decrypt and obtain the information; this works is fully dedicated to analyse the run time of the cipher method employing the Pseudoanalytic Function Theory. The main purpose of this work, is to analyse the cipher method exposed in [3] and calculate the time complexity in order to study the behaviour and look at the possibility to develop an optimized algorithm, preserving the property of confidentiality information, but decreasing the time lapse.

We study the Electrical Impedance Equation within a special class of domains, whose boundaries posses non-smooth points. The forward Dirichlet boundary value problem is solved bias a novel numerical method, based upon the Pseudoanalytic Function Theory, that does not require additional regularization techniques to fulfill the boundary condition at the non-smooth points.

Based upon the elements of the modern pseudoanalytic function theory, we analyze a new method for numerically solving the forward Dirichlet boundary value problem corresponding to the two-dimensional electrical impedance equation. The analysis is performed by introducing interpolating piecewise separable-variables conductivity functions in the unit circle. To warrant the effectiveness of the posed method, we consider several examples of conductivity functions, whose boundary conditions are exact solutions of the electrical impedance equation, performing a brief comparison with the finite element method. Finally, we discuss the possible contributions of these results to the field of the electrical impedance tomography.