Solving Poisson's Equation in Finite Difference Method with Matlab

• MATLAB
• indrani
In summary, The conversation is about a problem solving Poisson's equation using the finite difference method in MATLAB. The equation is written correctly as (δ^2 φ)/(δx^2 )=(-ρ/ϵ N(x)) in cartesian coordinates. N(x) is in the numerator and represents the doping concentration which is a Gaussian function. The equation is 1-D Poisson's equation with N(x) being a function of x only. The individual in the conversation suggests integrating both sides and applying boundary conditions to solve the problem.
indrani
I have a problem solving poisson equation in finite difference method using matlab.

the equation is (δ^2 φ)/(δx^2 )=(-ρ/ϵ N(x))

where N(x)= a1*exp(-((x-b1)/c1)^2) + a2*exp(-((x-b2)/c2)^2)

Is this written correctly?
$$\frac{\partial}{\partial x} \left( \frac{\partial \psi}{\partial x} \right)=\frac{- \rho}{\varepsilon} N\left( x \right)$$
Is this equation in cartesian coordinates, or spherical or cylindrical coordinates? Is N(x) in the numerator?

it is in cartesian coordinate.I have to solve in 1Dimention. N(x) is in numerator and it is the doping concentration which is a Gaussian in nature

Is this true (i.e., 1-D Poisson equation); with N(x) being a function of x only?
$$-d \left( \frac{\partial \psi}{\partial x} \right)= d E_x (x) =\frac{+ \rho}{\varepsilon} N\left( x \right) dx$$
Then can't you integrate both sides and put in boundary conditions?

1. What is Poisson's equation and why is it important in science?

Poisson's equation is a mathematical equation that describes the relationship between the distribution of electric charge in a region and the electric potential in that region. It is important in science because it is a fundamental equation in electromagnetism and is used to understand and model a wide range of physical phenomena, from electric fields and currents to gravity and fluid dynamics.

2. What is the Finite Difference Method and how does it relate to solving Poisson's equation?

The Finite Difference Method is a numerical technique for solving differential equations, including Poisson's equation. It involves dividing the region of interest into a grid of points and using discrete approximations to represent the derivatives in the equations. This allows us to convert the continuous differential equation into a system of algebraic equations that can be solved using computer algorithms.

3. What is Matlab and why is it commonly used for solving Poisson's equation with the Finite Difference Method?

Matlab is a high-level programming language and interactive environment that is widely used in scientific and engineering applications. It is commonly used for solving Poisson's equation with the Finite Difference Method because it has built-in functions and tools for handling matrices and solving systems of equations, making it well-suited for numerical calculations.

4. What are the steps involved in solving Poisson's equation in Matlab using the Finite Difference Method?

The steps involved in solving Poisson's equation in Matlab using the Finite Difference Method include defining the grid of points, setting boundary conditions, discretizing the equation, constructing a system of equations, solving the system, and post-processing the results. More specifically, the process involves creating a matrix of coefficients, creating a vector of known values, and using the backslash operator to solve the system.

5. Are there any limitations to using the Finite Difference Method with Matlab to solve Poisson's equation?

While the Finite Difference Method with Matlab is a powerful and widely used technique for solving Poisson's equation, it does have some limitations. These include the need for a structured grid of points, which may not accurately represent complex geometries, and the potential for numerical errors and instability if the grid is not properly chosen. Additionally, the method may become computationally expensive for large and highly nonlinear problems.

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