Finding the fundamental solution of the bi-axially symmetric Laplace–Beltrami equation using generalized shift operator

Many physical processes are described by partial differential equations. The relevance of this study is due to the need to solve applied problems of quantum mechanics, the theory of elasticity, and heat capacity. In this paper, an equation is considered that describes the field created by a contour with two axes of symmetry. The purpose of the study is to find a fundamental solution to this equation, which can later be used when solving boundary value problems.

Newton-PGSS and Its Improvement Method for Solving Nonlinear Systems with Saddle Point Jacobian Matrices

The preconditioned generalized shift-splitting (PGSS) iteration method is unconditionally convergent for solving saddle point problems with nonsymmetric coefficient matrices. By making use of the PGSS iteration as the inner solver for the Newton method, we establish a class of Newton-PGSS method for solving large sparse nonlinear system with nonsymmetric Jacobian matrices about saddle point problems. For the new presented method, we give the local convergence analysis and semilocal convergence analysis under Hölder condition, which is weaker than Lipschitz condition. In order to further raise the efficiency of the algorithm, we improve the method to obtain the modified Newton-PGSS and prove its local convergence. Furthermore, we compare our new methods with the Newton-RHSS method, which is a considerable method for solving large sparse nonlinear system with saddle point nonsymmetric Jacobian matrix, and the numerical results show the efficiency of our new method.

Solving the Stochastic Generation and Transmission Capacity Planning Problem Applied to Large-Scale Power Systems Using Generalized Shift-Factors

In this study, we successfully develop the transmission planning problem of large-scale power systems based on generalized shift-factors. These distribution factors produce a reduced solution space which does not need the voltage bus angles to model new transmission investments. The introduced formulation copes with the stochastic generation and transmission capacity expansion planning problem modeling the operational problem using a 24-hourly load behaviour. Results show that this formulation achieves an important reduction of decision variables and constraints in comparison with the classical disjunctive transmission planning methodology known as the Big M formulation without sacrificing optimality. We test both the introduced and the Big M formulations to find out convergence and time performance using a commercial solver. Finally, several test power systems and extensive computational experiments are conducted to assess the capacity planning methodology. Solving deterministic and stochastic problems, we demonstrate a prominent reduction in the solver simulation time especially with large-scale power systems.

Image Processing Operators Based on the Gyrator Transform: Generalized Shift, Convolution and Correlation

The gyrator transform (GT) is used for images processing in applications of light propagation. We propose new image processing operators based on the GT, these operators are: Generalized shift, convolution and correlation. The generalized shift is given by a simultaneous application of a spatial shift and a modulation by a pure linear phase term. The new operators of convolution and correlation are defined using the GT. All these image processing operators can be used in order to design and implement new optical image processing systems based on the GT. The sampling theorem for images whose resulting GT has finite support is developed and presented using the previously defined operators. Finally, we describe and show the results for an optical image encryption system using a nonlinear joint transform correlator and the proposed image processing operators based on the GT.