Inhomogeneous thermal conduction equations

1978 ◽  
Vol 56 (7) ◽  
pp. 928-935
Author(s):  
C. S. Lai
Keyword(s):  
Differential Equations ◽  
Thermal Conduction ◽  
Series Solution ◽  
Nonlinear Differential Equations ◽  
Three Dimensional ◽  
Incompressible Fluids ◽  
Self Similar Solution ◽  
Self Similar ◽  
A New Technique ◽  
The One

The method of self-similar solution of partial differential equations is applied to the one-, two-, and three-dimensional inhomogeneous thermal conduction equations with the thermometric conductivities χ ~ rmWn. Analytical solutions are obtained for the case that the total amount of heat is conserved. For the case that the temperature is maintained constant at r = 0, a new technique of the series solution about the point of intercept is proposed to solve the resultant nonlinear differential equations. The solutions obtained are useful in studying the thermal conduction characteristics of some incompressible fluids.

scholarly journals Approximate Solutions of Nonlinear Partial Differential Equations by Modifiedq-Homotopy Analysis Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Shaheed N. Huseen ◽  
Said R. Grace
Keyword(s):  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Series Solution ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Approximate Solutions ◽  
Power Series Solution ◽  
Analysis Method ◽  
Exactly Solvable ◽  
Homotopy Analysis

A modifiedq-homotopy analysis method (mq-HAM) was proposed for solvingnth-order nonlinear differential equations. This method improves the convergence of the series solution in thenHAM which was proposed in (see Hassan and El-Tawil 2011, 2012). The proposed method provides an approximate solution by rewriting thenth-order nonlinear differential equation in the form ofnfirst-order differential equations. The solution of thesendifferential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.

A self-similar solution for expansion into a vacuum of a collisionless plasma bunch

1998 ◽  
Vol 59 (1) ◽  
pp. 83-90 ◽  
Author(s):  
A. V. BAITIN ◽  
K. M. KUZANYAN
Keyword(s):  
Electron Distribution Function ◽  
Collisionless Plasma ◽  
Electron Component ◽  
One Dimensional ◽  
Ultrarelativistic Electron ◽  
Plasma Bunch ◽  
Self Similar Solution ◽  
Self Similar ◽  
The One ◽  
Similar Solutions

The process of expansion into a vacuum of a collisionless plasma bunch with relativistic electron temperature is investigated for the one-dimensional case. Self-similar solutions for the evolution of the electron distribution function and ion acceleration are obtained, taking account of cooling of the electron component of plasma for the cases of non-relativistic and ultrarelativistic electron energies.

Determination of Optimum Profile of One-Dimensional Cooling Fins

1983 ◽  
Vol 105 (3) ◽  
pp. 317-320 ◽  
Author(s):  
S. K. Hati ◽  
S. S. Rao
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Minimum Principle ◽  
Problem Formulation ◽  
One Dimensional ◽  
A New Technique ◽  
The One ◽  
Optimum Profile

The optimum design of an one-dimensional cooling fin is considered by including all modes of heat transfer in the problem formulation. The minimum principle of Pontryagin is applied to determine the optimum profile. A new technique is used to solve the reduced differential equations with split boundary conditions. The optimum profile found is compared with the one obtained by considering only conduction and convection.

Wavelet based method for solving generalized Burger’s type equations

2019 ◽  
Vol 08 (04) ◽  
pp. 1950020
Author(s):  
Inderdeep Singh
Keyword(s):  
Differential Equations ◽  
Algebraic System ◽  
Numerical Experiments ◽  
Nonlinear Differential Equations ◽  
Linear Equations ◽  
Nonlinear Partial Differential Equations ◽  
Three Dimensional ◽  
Haar Wavelet ◽  
Nonlinear Ordinary Differential Equation ◽  
System Of Linear Equations

In this work, an efficient numerical method is proposed for solving generalized Burger’s type equations. The generalized Burger’s type equations are first converted into a nonlinear ordinary differential equation by choosing some suitable wave variable transformation. Linearize such nonlinear differential equations by using quasilinearization technique. For solving algebraic system of linear equations Haar wavelet-based collocation method is used. A distinct feature of the proposed method is their simple applicability in a variety of two- and three- dimensional nonlinear partial differential equations. Numerical experiments are performed to illustrate the accuracy and efficiency of the proposed method.

scholarly journals Formation of corner waves in the wake of a partially submerged bluff body

2015 ◽  
Vol 771 ◽  
pp. 547-563 ◽  
Author(s):  
P. Martínez-Legazpi ◽  
J. Rodríguez-Rodríguez ◽  
A. Korobkin ◽  
J. C. Lasheras
Keyword(s):  
Bluff Body ◽  
Physical Mechanism ◽  
Essential Feature ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Point Of View ◽  
Similar Solution ◽  
Physical Point ◽  
Self Similar Solution ◽  
Self Similar

We study theoretically and numerically the downstream flow near the corner of a bluff body partially submerged at a deadrise depth ${\rm\Delta}h$ into a uniform stream of velocity $U$, in the presence of gravity, $g$. When the Froude number, $\mathit{Fr}=U/\sqrt{g{\rm\Delta}h}$, is large, a three-dimensional steady plunging wave, which is referred to as a corner wave, forms near the corner, developing downstream in a similar way to a two-dimensional plunging wave evolving in time. We have performed an asymptotic analysis of the flow near this corner to describe the wave’s initial evolution and to clarify the physical mechanism that leads to its formation. Using the two-dimensions-plus-time approximation, the problem reduces to one similar to dam-break flow with a wet bed in front of the dam. The analysis shows that, at leading order, the problem admits a self-similar formulation when the size of the wave is small compared with the height difference ${\rm\Delta}h$. The essential feature of the self-similar solution is the formation of a mushroom-shaped jet from which two smaller lateral jets stem. However, numerical simulations show that this self-similar solution is questionable from the physical point of view, as the two lateral jets plunge onto the free surface, leading to a self-intersecting flow. The physical mechanism leading to the formation of the mushroom-shaped structure is discussed.

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