# Deriving the derivative boundary conditions from natural formulation

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• maistral
In summary, the conversation discusses the use of a general derivative boundary condition and its relationship to the general natural boundary condition formulation. The main problem is that the left and right sides have different results, which is addressed by considering the heat flux in the x direction. The conversation also explains the equations for the fictitious points at the left and right boundaries.
maistral
TL;DR Summary
How to derive the finite difference derivative formulation from the natural boundary formulation?
PS: This is not an assignment, this is more of a brain exercise.

I intend to apply a general derivative boundary condition f(x,y). While I know that the boxed formulation is correct, I have no idea how to acquire the same formulation if I come from the general natural boundary condition formulation. I honestly do not know what am I doing wrong. Can someone check where am I incorrect?

exln
Looks OK. So what is the problem?

Chestermiller said:
Looks OK. So what is the problem?

The left side and the right side have different results.

If ##q_x## is the heat flux in the x direction, then ##q_x## is always given by $$q_x=-k\frac{dT}{dx}$$irrespective of whether it is the left- or the right boundary. Of course, the sign of the flux in the x direction can be negative. So, at the fictitious point at the left boundary, you have:
$$T(-\Delta x)=T(+\Delta x)+q_x(0)\Delta x$$and, at the fictitious point at the right boundary, you have:
$$T(L+\Delta x)=T(L-\Delta x)-q_x(L)\Delta x$$

## 1. What is the purpose of deriving the derivative boundary conditions from natural formulation?

The purpose of deriving the derivative boundary conditions from natural formulation is to accurately describe the behavior of a physical system at its boundaries. This helps in solving differential equations and understanding the physical processes involved.

## 2. How are the derivative boundary conditions derived from natural formulation?

The derivative boundary conditions are derived by considering the physical laws and principles that govern the behavior of the system at its boundaries. This involves using mathematical techniques such as the chain rule and integration by parts to express the derivatives in terms of the natural variables.

## 3. What are some examples of derivative boundary conditions derived from natural formulation?

Some examples of derivative boundary conditions derived from natural formulation include the heat equation, where the heat flux at the boundary is equal to the product of the thermal conductivity and the temperature gradient, and the Navier-Stokes equations, where the shear stress at the boundary is equal to the product of the dynamic viscosity and the velocity gradient.

## 4. Why is it important to consider derivative boundary conditions in scientific research?

Derivative boundary conditions are important in scientific research because they provide a more accurate representation of the behavior of a physical system at its boundaries. This can lead to more precise predictions and better understanding of the underlying processes involved.

## 5. Are there any limitations to deriving derivative boundary conditions from natural formulation?

Yes, there are limitations to deriving derivative boundary conditions from natural formulation. In some cases, it may be difficult to accurately describe the behavior of a physical system at its boundaries using only natural variables. Additionally, the derivation process may become complex and time-consuming for more complicated systems.

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