scholarly journals Finite Analytic Method for One-Dimensional Nonlinear Consolidation under Time-Dependent Loading

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Dawei Cheng ◽  
Wenke Wang ◽  
Xi Chen ◽  
Zaiyong Zhang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Time Dependent ◽  
Analytic Method ◽  
Partial Differential ◽  
One Dimensional ◽  
Finite Analytic Method ◽  
Nonlinear Consolidation

For one-dimensional (1D) nonlinear consolidation, the governing partial differential equation is nonlinear. This paper develops the finite analytic method (FAM) to simulate 1D nonlinear consolidation under different time-dependent loading and initial conditions. To achieve this, the assumption of constant initial effective stress is not considered and the governing partial differential equation is transformed into the diffusion equation. Then, the finite analytic implicit scheme is established. The convergence and stability of finite analytic numerical scheme are proven by a rigorous mathematical analysis. In addition, the paper obtains three corrected semianalytical solutions undergoing suddenly imposed constant loading, single ramp loading, and trapezoidal cyclic loading, respectively. Comparisons of the results of FAM with the three semianalytical solutions and the result of FDM, respectively, show that the FAM can obtain stable and accurate numerical solutions and ensure the convergence of spatial discretization for 1D nonlinear consolidation.

scholarly journals A Numerical Well-Balanced Scheme for One-Dimensional Heat Transfer in Longitudinal Triangular Fins

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
I. Rusagara ◽  
C. Harley
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Nonlinear Partial Differential Equation ◽  
Transfer Coefficients ◽  
Partial Differential ◽  
One Dimensional ◽  
Heat Transfer Coefficients ◽  
Highly Nonlinear ◽  
Triangular Fin

The temperature profile for fins with temperature-dependent thermal conductivity and heat transfer coefficients will be considered. Assuming such forms for these coefficients leads to a highly nonlinear partial differential equation (PDE) which cannot easily be solved analytically. We establish a numerical balance rule which can assist in getting a well-balanced numerical scheme. When coupled with the zero-flux condition, this scheme can be used to solve this nonlinear partial differential equation (PDE) modelling the temperature distribution in a one-dimensional longitudinal triangular fin without requiring any additional assumptions or simplifications of the fin profile.

scholarly journals Numerical Solution of European and American Option with Dividends using Finite Difference Methods

2020 ◽  
Vol 13 (13) ◽  
pp. 55-61
Author(s):  
Kedar Nath Uprety ◽  
Ganesh Prasad Panday
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Analytical Formula ◽  
Partial Differential ◽  
Dividend Payments ◽  
Fully Implicit ◽  
Black Scholes ◽  
Nicolson Scheme

Numerical methods form an important part of the pricing of financial derivatives where there is no closed form analytical formula. Black-Scholes equation is a well known partial differential equation in financial mathematics. In this paper, we have studied the numerical solutions of the Black-Scholes equation for European options (Call and Put) as well as American options with dividends. We have used different approximate to discretize the partial differential equation in space and explicit (Forward Euler’s), fully implicit with projected Successive Over-Relaxation (SOR) algorithm and Crank-Nicolson scheme for time stepping. We have implemented and tested the methods in MATLAB. Finally, some numerical results have been presented and the effects of dividend payments on option pricing have also been considered.

Unsteady Flow of Gas Through a Semi-Infinite Porous Medium

1957 ◽  
Vol 24 (3) ◽  
pp. 329-332
Author(s):  
R. E. Kidder
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Porous Medium ◽  
Gas Flow ◽  
Transient Flow ◽  
Order Term ◽  
Nonlinear Partial Differential Equation ◽  
Flow In Porous Media ◽  
Partial Differential ◽  
One Dimensional

Abstract This paper presents an analytic solution to a problem of the transient flow of gas within a one-dimensional semi-infinite porous medium. A perturbation method, carried out to include terms of the second order, is employed to obtain a solution of the nonlinear partial differential equation describing the flow of gas. The zero-order term of the solution represents the solution of the linearized partial differential equation of gas flow in porous media given by Green and Wilts (1).

SYSTEM CHARACTERISTICS OF DISTRIBUTED PARAMETER SYSTEMS

2020 ◽  
Vol 70 (3) ◽  
pp. 34-44
Author(s):  
Kamen Perev
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Initial Conditions ◽  
Distributed Parameter Systems ◽  
Point Of View ◽  
Distributed Parameter ◽  
Partial Differential ◽  
Special Cases ◽  
The Green Function ◽  
Set Up

The paper considers the problem of distributed parameter systems modeling. The basic model types are presented, depending on the partial differential equation, which determines the physical processes dynamics. The similarities and the differences with the models described in terms of ordinary differential equations are discussed. A special attention is paid to the problem of heat flow in a rod. The problem set up is demonstrated and the methods of its solution are discussed. The main characteristics from a system point of view are presented, namely the Green function and the transfer function. Different special cases for these characteristics are discussed, depending on the specific partial differential equation, as well as the initial conditions and the boundary conditions.

Similarity solutions and viscous gravity current adjustment times

2019 ◽  
Vol 874 ◽  
pp. 285-298 ◽  
Author(s):  
Thomasina V. Ball ◽  
Herbert E. Huppert
Keyword(s):  
Differential Equation ◽  
Similarity Solution ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Gravity Current ◽  
Similarity Solutions ◽  
Partial Differential ◽  
Initial Shape ◽  
Wide Range

A wide range of initial-value problems in fluid mechanics in particular, and in the physical sciences in general, are described by nonlinear partial differential equations. Recourse must often be made to numerical solutions, but a powerful, well-established technique is to solve the problem in terms of similarity variables. A disadvantage of the similarity solution is that it is almost always independent of any specific initial conditions, with the solution to the full differential equation approaching the similarity solution for times $t\gg t_{\ast }$, for some $t_{\ast }$. But what is $t_{\ast }$? In this paper we consider the situation of viscous gravity currents and obtain useful formulae for the time of approach, $\unicode[STIX]{x1D70F}(p)$, for a number of different initial shapes, where $p$ is the percentage disagreement between the radius of the current as determined by the full numerical solution of the governing partial differential equation and the similarity solution normalised by the similarity solution. We show that for any initial shape of volume $V,\unicode[STIX]{x1D70F}\propto 1/(\unicode[STIX]{x1D6FD}V^{1/3}\unicode[STIX]{x1D6FE}_{0}^{8/3}p)$ (as $p\downarrow 0$), where $\unicode[STIX]{x1D6FD}=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/(3\unicode[STIX]{x1D707})$, with $g$ representing the acceleration due to gravity, $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ the density difference between the gravity current and the ambient, $\unicode[STIX]{x1D707}$ the dynamic viscosity of the fluid that makes up the gravity current and $\unicode[STIX]{x1D6FE}_{0}$ the initial aspect ratio. This framework can used in many other situations, including where it is not an initial condition (in time) that is studied but one valid for specified values at a special spatial coordinate.

Uniqueness of the null solution to a nonlinear partial differential equation satisfied by the explosion probability of a branching diffusion

2016 ◽  
Vol 53 (3) ◽  
pp. 938-945 ◽  
Author(s):  
K. Bruce Erickson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equation ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Null Solution ◽  
Nonlinear Parabolic ◽  
Branching Diffusion ◽  
Creation Rate

AbstractThe explosion probability before time t of a branching diffusion satisfies a nonlinear parabolic partial differential equation. This equation, along with the natural boundary and initial conditions, has only the trivial solution, i.e. explosion in finite time does not occur, provided the creation rate does not grow faster than the square power at ∞.

Sign in / Sign up

Export Citation Format

Share Document