Solving Poison's Equation - Method Explanation

  • Thread starter GarageDweller
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In summary, The conversation discusses a method for solving poisons equation by factoring out either P(x) or A(y) from the left hand side, and then obtaining a pair of constants for G''(x) and G(x). However, this method only works for certain functions P(x).
  • #1
GarageDweller
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I thought of a method to solving poisons equation the other day, tell me if this is right.

we have, ∂(∂ψ/∂x)/∂x+∂(∂ψ/∂y)/∂y=P(x)A(y)
Assuming the solution is the product of two single variable functions,

G''(x)V(y)+V''(y)G(x)=P(x)A(y)

However, for the left hand side to equal the right hand side, we must be able to factor out either P(x) or A(y) from both terms, for the sake of this explanation, we will take P(x) to be the one that could be factored out, dividing by P(x) on both sides..

(G''(x)/P(x))V(y)+(G(x)/P(x))V''(y)=A(y)

Now from this, we can see that G''(x)/P(x) and G(x)/P(x) equal some pair of constants (k,q)

kV(y)+qV''(y)=A(y)
G''(x)+G(x)=(k+q)P(x)
 
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  • #2
GarageDweller said:
(G''(x)/P(x))V(y)+(G(x)/P(x))V''(y)=A(y)

Now from this, we can see that G''(x)/P(x) and G(x)/P(x) equal some pair of constants (k,q)

kV(y)+qV''(y)=A(y)
G''(x)+G(x)=(k+q)P(x)
If G''(x)/P(x) and G(x)/P(x) equal the pair of constants (k,q) then G''(x) = kP(x), G(x) = qP(x). Thus qP''(x) = kP(x). So this is only going to work for certain functions P(x).
 

1. What is Poison's Equation?

Poison's Equation, also known as the Poison Distribution or Poison Process, is a mathematical equation that describes the probability of a given number of events occurring in a fixed time interval, if these events occur with a known average rate and independently of the time since the last event.

2. Why is it important to solve Poison's Equation?

Solving Poison's Equation is important because it allows us to accurately model and predict the behavior of processes that involve discrete events occurring randomly over time. This has applications in various fields such as biology, physics, and economics.

3. What is the method used to solve Poison's Equation?

The most commonly used method to solve Poison's Equation is the Poisson Distribution, which is a discrete probability distribution that gives the probability of a certain number of events occurring in a fixed interval of time. This method involves calculating the mean and using it to determine the probability of each possible number of events.

4. What are the assumptions made when using Poison's Equation?

The main assumptions made when using Poison's Equation include that the events occur independently of each other, the average rate of events is constant, and the probability of an event occurring is the same for all time intervals of equal length.

5. How is Poison's Equation applied in real-world situations?

Poison's Equation is applied in a variety of real-world situations, such as predicting the number of customers arriving at a store during a certain time period, estimating the number of defects in a manufacturing process, and modeling the spread of diseases in a population. It can also be used in risk assessment and insurance calculations.

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