Numerical solutions of an integro-differential equation with smooth and singular kernels

2019 ◽  
Vol 13 (12) ◽  
pp. 573-586
Author(s):  
F. M. Alharbi
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Integro Differential Equation ◽  
Singular Kernels

II. A solution of the integro-differential equation for the slope of the jet

1961 ◽  
Vol 261 (1304) ◽  
pp. 97-118 ◽  
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Coordinate Transformation ◽  
Present Author ◽  
Numerical Solutions ◽  
Two Dimensional ◽  
Integro Differential Equation ◽  
Mellin Transforms ◽  
Momentum Coefficient ◽  
Slope Function

In an earlier paper with the same general title (Spence 1956, referred to as I), a mathematical model was developed to discuss the flow past a two-dimensional wing at incidence a in a steady incompressible stream, with a jet of momentum coefficient C j emerging from the trailing edge at an angular deflexion r to the chordline. In linearized approximation it was shown that the slope of the jet is given by a certain singular integro-differential equation, and numerical solutions for the equation were obtained by a pivotal points method. A coordinate transformation has now been found (Spence 1959) which makes the equation independent of the jet strength for small values of 14 C j = u, say, yielding a simpler equation solved by Lighthill (1959) using Mellin transforms (and by Stewartson (1959) and the present author by other methods). In this paper the expansion of the slope function is continued in ascending powers of u and In u multiplied by functions of x found by solving, by Lighthill’s method, a series of closely-related inhomogeneous equations. From these, expansions of the lift derivatives with respect to a and r are found as To this order the expressions agree closely with the numerical results found earlier, the discrepancy at u = 1 being less than 4 %.

scholarly journals A New Integro-Differential Equation for Rossby Solitary Waves with Topography Effect in Deep Rotational Fluids

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Hongwei Yang ◽  
Qingfeng Zhao ◽  
Baoshu Yin ◽  
Huanhe Dong
Keyword(s):  
Differential Equation ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Pseudospectral Method ◽  
Multiple Scale ◽  
Integro Differential Equation ◽  
Topography Effect ◽  
Detuning Parameter ◽  
Rossby Solitary Waves

From rotational potential vorticity-conserved equation with topography effect and dissipation effect, with the help of the multiple-scale method, a new integro-differential equation is constructed to describe the Rossby solitary waves in deep rotational fluids. By analyzing the equation, some conservation laws associated with Rossby solitary waves are derived. Finally, by seeking the numerical solutions of the equation with the pseudospectral method, by virtue of waterfall plots, the effect of detuning parameter and dissipation on Rossby solitary waves generated by topography are discussed, and the equation is compared with KdV equation and BO equation. The results show that the detuning parameterαplays an important role for the evolution features of solitary waves generated by topography, especially in the resonant case; a large amplitude nonstationary disturbance is generated in the forcing region. This condition may explain the blocking phenomenon which exists in the atmosphere and ocean and generated by topographic forcing.

scholarly journals The validity of the kinetic collection equation revisited – Part 2: Simulations for the hydrodynamic kernel

2010 ◽  
Vol 10 (3) ◽  
pp. 6219-6240
Author(s):  
L. Alfonso ◽  
G. B. Raga ◽  
D. Baumgardner
Keyword(s):  
Differential Equation ◽  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Numerical Solution ◽  
Mass Distribution ◽  
Total Mass ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Integro Differential Equation ◽  
Simulation Results

Abstract. The kinetic collection equation (KCE) has been widely used to describe the evolution of the average droplet spectrum due to the collection process that leads to the development of precipitation in warm clouds. This deterministic, integro-differential equation only has analytic solution for very simple kernels. For more realistic kernels, the KCE needs to be integrated numerically. In this study, the validity time of the KCE for the hydrodynamic kernel is estimated by a direct comparison of Monte Carlo simulations with numerical solutions of the KCE. The simulation results show that when the largest droplet becomes separated from the smooth spectrum, the total mass calculated from the numerical solution of the KCE is not conserved and, thus, the KCE is no longer valid. This result confirms the fact that for realistic kernels appropriate for precipitation development within warm clouds, the KCE can only be applied to the continuous portion of the mass distribution.

scholarly journals Self-similar fault slip in response to fluid injection

2021 ◽  
Vol 928 ◽  
Author(s):  
Robert C. Viesca
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Fluid Pressure ◽  
Perturbation Expansion ◽  
Fault Slip ◽  
Length Scale ◽  
Matched Asymptotics ◽  
Integro Differential Equation ◽  
Simple Boundary ◽  
Self Similar

There is scientific and industrial interest in understanding how geologic faults respond to transient sources of fluid. Natural and artificial sources can elevate pore fluid pressure on the fault frictional interface, which may induce slip. We consider a simple boundary value problem to provide an elementary model of the physical process and to provide a benchmark for numerical solution procedures. We examine the slip of a fault that is an interface of two elastic half-spaces. Injection is modelled as a line source at constant pressure and fluid pressure is assumed to diffuse along the interface. The resulting problem is an integro-differential equation governing fault slip, which has a single dimensionless parameter. The expansion of slip is self-similar and the rupture front propagates at a factor $\lambda$ of the diffusive length scale $\sqrt {\alpha t}$ . We identify two asymptotic regimes corresponding to $\lambda$ being small or large and perform a perturbation expansion in each limit. For large $\lambda$ , in the regime of a so-called critically stressed fault, a boundary layer emerges on the diffusive length scale, which lags far behind the rupture front. We demonstrate higher-order matched asymptotics for the integro-differential equation, and in doing so, we derive a multipole expansion to capture successive orders of influence on the outer problem for fault slip for a driving force that is small relative to the crack dimensions. Asymptotic expansions are compared with accurate numerical solutions to the full problem, which are tabulated to high precision.

scholarly journals The validity of the kinetic collection equation revisited – Part 2: Simulations for the hydrodynamic kernel

2010 ◽  
Vol 10 (15) ◽  
pp. 7189-7195 ◽  
Author(s):  
L. Alfonso ◽  
G. B. Raga ◽  
D. Baumgardner
Keyword(s):  
Differential Equation ◽  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Numerical Solution ◽  
Mass Distribution ◽  
Total Mass ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Integro Differential Equation ◽  
Simulation Results

Abstract. The kinetic collection equation (KCE) has been widely used to describe the evolution of the average droplet spectrum due to the collection process that leads to the development of precipitation in warm clouds. This deterministic, integro-differential equation only has analytic solution for very simple kernels. For more realistic kernels, the KCE needs to be integrated numerically. In this study, the validity time of the KCE for the hydrodynamic kernel is estimated by a direct comparison of Monte Carlo simulations with numerical solutions of the KCE. The simulation results show that when the largest droplet becomes separated from the smooth spectrum, the total mass calculated from the numerical solution of the KCE is not conserved and, thus, the KCE is no longer valid. This result confirms the fact that for kernels appropriate for precipitation development within warm clouds, the KCE can only be applied to the continuous portion of the mass distribution.

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