An initial value problem for a class of higher order partial differential equations related to the heat equation

1973 ◽  
Vol 97 (1) ◽  
pp. 115-187 ◽  
Author(s):  
J. B. Diaz ◽  
Curtis S. Means
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Heat Equation ◽  
Initial Value Problem ◽  
Higher Order ◽  
Initial Value ◽  
Partial Differential

The initial-value problem for long waves of finite amplitude

1964 ◽  
Vol 20 (1) ◽  
pp. 161-170 ◽  
Author(s):  
Robert R. Long
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Integration ◽  
Initial Value Problem ◽  
Finite Amplitude ◽  
Long Waves ◽  
Initial Value ◽  
Partial Differential ◽  
Long Wave ◽  
Constant Slope

Derived herein is a set of partial differential equations governing the propagation of an arbitrary, long-wave disturbance of small, but finite amplitude. The equations reduce to that of Boussinesq (1872) when the assumption is made that the disturbance is propagating in one direction only. The equations are hyperbolic with characteristic curves of constant slope. The initial-value problem can be solved very readily by numerical integration along characteristics. A few examples are included.

scholarly journals Analysis of Different Boundary Conditions on Homogeneous One-Dimensional Heat Equation

2021 ◽  
Vol 7 (1) ◽  
pp. 15-21
Author(s):  
Norazlina Subani ◽  
Muhammad Aniq Qayyum Mohamad Sukry ◽  
Muhammad Arif Hannan ◽  
Faizzuddin Jamaluddin ◽  
Ahmad Danial Hidayatullah Badrolhisam
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Initial Value ◽  
Partial Differential ◽  
Different Boundary Conditions ◽  
One Dimensional ◽  
Profile Changes ◽  
The Right

Partial differential equations involve results of unknown functions when there are multiple independent variables. There is a need for analytical solutions to ensure partial differential equations could be solved accurately. Thus, these partial differential equations could be solved using the right initial and boundaries conditions. In this light, boundary conditions depend on the general solution; the partial differential equations should present particular solutions when paired with varied boundary conditions. This study analysed the use of variable separation to provide an analytical solution of the homogeneous, one-dimensional heat equation. This study is applied to varied boundary conditions to examine the flow attributes of the heat equation. The solution is verified through different boundary conditions: Dirichlet, Neumann, and mixed-insulated boundary conditions. the initial value was kept constant despite the varied boundary conditions. There are two significant findings in this study. First, the temperature profile changes are influenced by the boundary conditions, and that the boundary conditions are dependent on the heat equation’s flow attributes.

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