Finite Element Method (FEM), when applied to solve problems, has faced some challenges over the years, such as time consumption and the complexity of assumptions. In particular, the making of assumptions has had a sig...Finite Element Method (FEM), when applied to solve problems, has faced some challenges over the years, such as time consumption and the complexity of assumptions. In particular, the making of assumptions has had a significant influence on the accuracy of the method, making it mandatory to carry out sensitivity analysis. The sensitivity analysis helps to identify the level of impact the assumptions have on the method. However, sensitivity analysis via FEM can be very challenging. A priori error estimation, an integral part of FEM, is a basic mathematical tool for predicting the accuracy of numerical solutions. By understanding the relationship between the mesh size, the order of basis functions, and the resulting error, practitioners can effectively design and apply FEM to solve complex Partial Differential Equations (PDEs) with confidence in the reliability of their results. Thus, the coercive property and A priori error estimation based on the L1 formula on a mesh in time and the Mamadu-Njoseh basis functions in space are investigated for a linearly distributed time-order fractional telegraph equation with restricted initial conditions. For this purpose, we constructed a mathematical proof of the coercive property for the fully discretized scheme. Also, we stated and proved a cardinal theorem for a priori error estimation of the approximate solution for the fully discretized scheme. We noticed the role of the restricted initial conditions imposed on the solution in the analysis of a priori error estimation.展开更多
It is generally known that the solutions of deterministic and stochastic differential equations （SDEs） usually grow linearly at such a rate that they may become unbounded after a small lapse of time and may eventual...It is generally known that the solutions of deterministic and stochastic differential equations （SDEs） usually grow linearly at such a rate that they may become unbounded after a small lapse of time and may eventually blow up or explode in finite time. If the drift and diffusion functions are globally Lipschitz, linear growth may still be experienced, as well as a possible blow-up of solutions in finite time. In this paper, a nonlinear scalar delay differential equation with a constant time lag is perturbed by a multiplicative Ito-type time - space white noise to form a stochastic Fokker-Planck delay differential equation. It is established that no explosion is possible in the presence of any intrinsically slow time - space white noise of Ito - type as manifested in the resulting stochastic Fokker- Planck delay differential equation. Time - space white noise has a role to play since the solution of the classical nonlinear equation without it still exhibits explosion.展开更多
文摘Finite Element Method (FEM), when applied to solve problems, has faced some challenges over the years, such as time consumption and the complexity of assumptions. In particular, the making of assumptions has had a significant influence on the accuracy of the method, making it mandatory to carry out sensitivity analysis. The sensitivity analysis helps to identify the level of impact the assumptions have on the method. However, sensitivity analysis via FEM can be very challenging. A priori error estimation, an integral part of FEM, is a basic mathematical tool for predicting the accuracy of numerical solutions. By understanding the relationship between the mesh size, the order of basis functions, and the resulting error, practitioners can effectively design and apply FEM to solve complex Partial Differential Equations (PDEs) with confidence in the reliability of their results. Thus, the coercive property and A priori error estimation based on the L1 formula on a mesh in time and the Mamadu-Njoseh basis functions in space are investigated for a linearly distributed time-order fractional telegraph equation with restricted initial conditions. For this purpose, we constructed a mathematical proof of the coercive property for the fully discretized scheme. Also, we stated and proved a cardinal theorem for a priori error estimation of the approximate solution for the fully discretized scheme. We noticed the role of the restricted initial conditions imposed on the solution in the analysis of a priori error estimation.
文摘It is generally known that the solutions of deterministic and stochastic differential equations （SDEs） usually grow linearly at such a rate that they may become unbounded after a small lapse of time and may eventually blow up or explode in finite time. If the drift and diffusion functions are globally Lipschitz, linear growth may still be experienced, as well as a possible blow-up of solutions in finite time. In this paper, a nonlinear scalar delay differential equation with a constant time lag is perturbed by a multiplicative Ito-type time - space white noise to form a stochastic Fokker-Planck delay differential equation. It is established that no explosion is possible in the presence of any intrinsically slow time - space white noise of Ito - type as manifested in the resulting stochastic Fokker- Planck delay differential equation. Time - space white noise has a role to play since the solution of the classical nonlinear equation without it still exhibits explosion.
