# PDE: Initial Conditions Contradicting Boundary Conditions

• Arkuski
In summary, the given IBVP has a PDE of u_{t}=α^{2}u_{xx} on the domain 0<x<1 and 0<t<∞ with boundary conditions u(0,t)=0 and u_{x}(1,t)=1 for all t, and an initial condition of u(x,0)=sin(πx) for 0≤x≤1. There may appear to be a mismatch between the boundary conditions and the initial condition, but it can be reconciled by using a substitution to make the system homogeneous.
Arkuski
Suppose we have the following IBVP:

PDE: $u_{t}=α^{2}u_{xx}$ $0<x<1$ $0<t<∞$
BCs: $u(0,t)=0, u_{x}(1,t)=1$ $0<t<∞$
IC: $u(x,0)=sin(πx)$ $0≤x≤1$

It appears as though the BCs and the IC do not match. The derivative of temperature with respect to x at position x=1 is a constant 1 while with the initial condition, the derivative is equal to -π. Do I conclude that the problem is incorrect or is there another way to reconcile this error?

I don't think there is an error. You can think of a bar with the given temperature distribution inserted into a situation with those bc's. I would begin by using a substitution like ##u(x,t) = v(x,t)+\psi(x)## to make a homogeneous system in ##v(x,t)## and let the ##\psi(x)## take care of the ##u_x(1,t)=1## nonhomogeneous term.

## 1. What is the meaning of PDE?

PDE stands for Partial Differential Equation. It is a type of mathematical equation that involves multiple variables and their partial derivatives. PDEs are commonly used in physics, engineering, and other fields to describe complex systems and phenomena.

## 2. What are initial conditions and boundary conditions in PDEs?

In PDEs, initial conditions refer to the values of the dependent variables at the starting point of the system. On the other hand, boundary conditions refer to the values of the dependent variables at the edges or boundaries of the system. These conditions are necessary to solve a PDE and obtain a unique solution.

## 3. How can initial conditions contradict boundary conditions in PDEs?

In some cases, the initial conditions and boundary conditions may not be compatible with each other. This means that the values of the dependent variables at the starting point of the system may not match the values at the edges or boundaries. This can lead to a contradiction, making it impossible to find a solution that satisfies both conditions.

## 4. Why is it important to ensure that initial conditions and boundary conditions are consistent in PDEs?

Consistency between initial conditions and boundary conditions is crucial in PDEs because it ensures that the solution obtained is valid and physically meaningful. If there is a contradiction, the solution may not accurately represent the behavior of the system, leading to incorrect predictions and conclusions.

## 5. How can one resolve the issue of initial conditions contradicting boundary conditions in PDEs?

There are several ways to resolve this issue, depending on the specific problem. One approach is to re-examine the physical assumptions and mathematical model used to formulate the PDE. Another approach is to adjust the initial or boundary conditions to make them more consistent. In some cases, it may also be necessary to use different numerical methods or techniques to solve the PDE.

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