Modified HPMs Inspired by Homotopy Continuation Methods

Nonlinear differential equations have applications in the modelling area for a broad variety of phenomena and physical processes; having applications for all areas in science and engineering. At the present time, the homotopy perturbation method (HPM) is amply used to solve in an approximate or exact manner such nonlinear differential equations. This method has found wide acceptance for its versatility and ease of use. The origin of the HPM is found in the coupling of homotopy methods with perturbation methods. Homotopy methods are a well established research area with applications, in particular, an applied branch of such methods are the homotopy continuation methods, which are employed on the numerical solution of nonlinear algebraic equation systems. Therefore, this paper presents two modified versions of standard HPM method inspired in homotopy continuation methods. Both modified HPMs deal with nonlinearities distribution of the nonlinear differential equation. Besides, we will use a calcium-induced calcium released mechanism model as study case to test the proposed techniques. Finally, results will be discussed and possible research lines will be proposed using this work as a starting point

Estimation of flood design using the instantaneous hydrograph unit: a case study of the Tecolutla river watershed, Mexico

In this research an instantaneous hydrograph unit (IUh), is used to characterize the flood design at the El Remolino station, located in the Tecolutla river watershed. Rainfall data from 24 stations, all of them located within, or in the vicinity of the watershed was used for the period 1961–1999 to learn about the flood design. From this period we selected 16 more intense storms recorded in the watershed, which generated a peak discharge expenditure greater than 3000 m3/s. The design hydrograph was obtained, using the maximum rainfall within 24 hours, for the return periods of 25, 50 and 100 years. It was found that the design hydrographs for 25 years would have a discharge peak of 21 053.7 m3/s, to 50 years of 23 653.5 m3/s and 100 years of 26 281.4 m3/s. These results indicate that the HUI has an over–estimation in the peak discharge, this over estimation might be because the design storms used have a duration in excess of four days and rainfall in excess of 495.4 mm, 557.2 mm and 618.6 mm for each return period. Finally, the peak discharge estimated, with the HUI, and the peak discharge registered, show a coefficient R2 = 0. 956, what shows the effectiveness of the model here proposed

Método de perturbación con transformada de Laplace para resolver problemas no lineales de múltiples soluciones, con condiciones a la frontera mixtas y Neumann

The field of differential equations has recently gained attention due to recent developments in science and technology. For this reason, the analysis for the use of new methodologies to solve them has become important. Based on the combination of Laplace Transform method (LT) and Perturbation Method (PM) this article proposes the Laplace transform-Perturbation Method (LT-PM) which finds its motivation on the application of LT to linear ordinary differential equations. The goal of this work is to propose a modification of PM - the LT-PM), in order to solve nonlinear perturbative problems with boundary conditions defined on finite intervals. The proposed methodology consisted on the application of LT to the differential equation to solve and then, assuming that its solutions can be expressed as a series of perturbative parameter powers. Thus, the solution of the problem is obtained by systematically applying the transformed inverse LT. The main results of this paper were shown through two case studies, where LT-PM is identified as potentially useful for finding multiple solutions to nonlinear problems. Additionally, the LT-PM enhances the applicability of PM, in some cases of mixed and Neumann boundary conditions, where PM is unsuitable to provide the results. With the purpose of verifying the accuracy of the obtained results, the Square Residual Error (SRE) was calculated. The resulting value was extremely low, which showed the precision and potential of LT-PM. We conclude that, although the proposed method resulted efficient for the case studies presented in this article, it is expected that LT-PM can be a potentially useful tool for other case studies. Particularly those related to the practical applications of science and engineering. El campo de las ecuaciones diferenciales ha cobrado auge en la actualidad por el desarrollo científico y tecnológico. Por esta situación, el estudio de nuevas metodologías para solucionarlas se ha vuelto importante. A partir de la combinación del método de Laplace Transform (LT) y el método de perturbación (PM) este trabajo presenta el método LT-PM, y su motivación se encuentra en la aplicación conocida de la LT a ecuaciones diferenciales ordinarias lineales. El objetivo de este trabajo fue presentar una modificación del método de perturbación (PM), el método de perturbación con transformada de Laplace (LT-PM), con el fin de resolver problemas perturbativos no lineales, con condiciones a la frontera definidas en intervalos finitos. La metodología consistió en aplicar LT a la ecuación diferencial por resolver y después de asumir que la solución de la misma se puede expresar como una serie de potencias de un parámetro perturbativo, se obtiene la solución del problema aplicando sistemáticamente la transformada inversa de Laplace. Los principales resultados de este trabajo se muestran a partir de dos casos de estudio presentados, donde se observa que LT-PM es potencialmente útil para encontrar soluciones múltiples de problemas no lineales. Además, LT-PM mejora la aplicabilidad del método de perturbación en algunos casos de condiciones a la frontera mixtas y de Neumann, donde PM simplemente no funciona. Con el fin de verificar la exactitud de los resultados obtenidos, se calculó su error residual cuadrático (SRE), el cual resultó muy bajo, de donde se dedujo su precisión y la potencialidad de LT-PM. Se concluye que si bien el método propuesto resulta eficiente en los casos particulares presentados, se espera que sea una herramienta potencialmente eficiente y útil para otros casos de estudio, particularmente, en aquellos relacionados con aplicaciones prácticas en ciencias e ingeniería