Multi-Dimensional Uncertain Calculus with Liu Process

Abstract Uncertain calculus deals with the integral and differential of some uncertain processes. So far, uncertain integrals have been defined with respect to Liu process, renewal process, finite variation process, and multiple Liu processes. This paper presents an uncertain integral of a matrix of uncertain processes with respect to multi-dimensional Liu process, and verifies its linearity property. Then an uncertain differential of a multi-dimensional uncertain process with respect to a multi-dimensional Liu process is defined, and a fundamental theorem is derived. In addition, a concept of multi-dimensional uncertain differential equation is proposed, and solutions of some special types of multi-dimensional uncertain differential equations are given

Fractional uncertain differential equations with general memory effects: Existences and alpha-path solutions

General fractional calculus is popular recently. Fractional uncertain differential equations (FUDEs) with general memory effects are proposed in this paper. Firstly, existence and uniqueness theorems of solution for general fractional uncertain differential equations (GFUDEs) is presented, and the analytic solution of a linear one is given. Then the concept of α-path is introduced, and relationship between solution of GFUDE and corresponding α-path is also discussed. In addition, a theorem is proved to obtain the expected value of a monotonic function related to solutions of GFUDEs. Finally, a numerical example is given to better understand the significance of general memory effects. This paper provides more types of FUDEs to better describe some phenomena in uncertain environments

Implementación del método de Crank—Nicolson para resolver la ecuación unidimensional del calor con incertidumbre para estudiantes de cuarto semestre, carrera de Telecomunicaciones de la Escuela Superior Politécnica de Chimborazo.

El objetivo fue implementar el método de Crank-Nicolson para la resolución de la ecuación unidimensional del calor con incertidumbre y mostrar este desarrollo a estudiantes de cuarto semestre de la Carrera de Telecomunicaciones de la Escuela Superior Politécnica de Chimborazo. Metodológicamente se realizó́ un análisis de los métodos de las diferencias finitas, de los elementos finitos y se puso de relieve al método de Euler, como una simplificación, y como un caso particular del método de diferencias finitas. Adicionalmente, se analizó́ al método de Crank-Nicolson y se lo presentó como una opción viable para resolver el problema, resolviendo varios ejercicios de aplicación. Se profundizó en el análisis del método de diferencias finitas para que este estudio sirva de fundamento para la posterior comparación de las eficiencias de ambos métodos. Se realizó el análisis de errores concomitantes a la aplicación de los métodos numéricos y se analizó́ la estabilidad de los métodos. Además, se utilizaron como parámetros de comparación el tiempo necesario para el cálculo y el costo computacional de la implementación de los métodos numéricos. Se realizó la difusión de resultados y la socialización de la aplicación del método con estudiantes de cuarto semestre de la Carrera de Telecomunicaciones de la Escuela Superior Politécnica de Chimborazo. Se presentaron los resultados de la encuesta de satisfacción planteada al mencionado grupo de estudiantes y se demostró́ la validez estadística de los resultados, por medio del cálculo del Alfa de Cronbach y de un análisis estadístico aplicando el software libre R.The objective was to implement the Crank-Nicolson method to resolve the one-dimensional heat equation with uncertainty and show this development to fourth-semester students of the Telecommunications Career of the Polytechnic School of Chimborazo. Methodologically, an analysis of the finite difference methods of the finite elements was carried out, and Euler's method was highlighted as a simplification and as a particular case of the finite difference method. Additionally, the Crank-Nicolson method was analyzed and presented as a viable option to solve the problem, solving several application exercises. The analysis of the finite difference method was deepened so that this study serves as a foundation for the subsequent comparison of the efficiencies of both methods. The analysis of errors concomitant to the application of the numerical methods was carried out, and the stability of the methods was analyzed. In addition, the time required for the calculation and the computational cost of implementing the numerical methods were used as comparison parameters. The dissemination of results and the socialization of the application of the method with fourth-semester students of the Telecommunications Career of the Higher Polytechnic School of Chimborazo were carried out. The results of the satisfaction survey posed to the group mentioned above students were presented, and the statistical validity of the results was demonstrated using the calculation of Cronbach's Alpha and a statistical analysis applying the free software R