Derive the finite difference scheme for the numerical solution of the first-order diffusion equation IBVP using the Crank-Nicolson method

. In the article, a differential scheme is created for the the first-order diffusion equation using the Crank-Nicolson method. The stability of the differential scheme was checked using the Neumann method. To solve the problem numerically, stability intervals were found using the Neman method. This work presents an analysis of the stability of the Crank-Nicolson scheme for the two-dimensional diffusion equation using Von Neumann stability analysis. The Crank-Nicolson scheme is a widely used numerical method for solving partial differential equations that combines the explicit and implicit schemes. The stability analysis is an important factor to consider when choosing a numerical method for solving partial differential equations, as numerical instability can cause inaccurate solutions. We show that the Crank-Nicolson scheme is unconditionally stable, meaning that it can be used for a wide range of parameters without being affected by numerical instability. Overall, the analysis and implementation presented in this work provide a framework for designing and analyzing numerical methods for solving partial differential equations using the Crank-Nicolson scheme. The stability analysis is crucial for ensuring the accuracy and reliability of numerical solutions of partial differential equations.


Introduction
The Crank-Nicolson method is a numerical technique for solving time-dependent partial differential equations (PDEs). It is a popular method for solving PDEs because it is unconditionally stable and has second-order accuracy in time and space. The method is based on the idea of taking the average of the solutions at two time levels, and using this average to approximate the solution at the midpoint between the two time levels. The Crank-Nicolson method can be applied to the first-order diffusion equation with initial and boundary conditions (IBVP): The Diffusion equation is Subject to the initial condition ( , 0) = ( ) and boundary conditions (0, ) = ( ) and ( , ) = ( ). Using the Crank-Nicolson method as described in the previous answer, we obtain the finite difference equation: 1st-order forward differences For a differential equation, we create an approximating differential scheme

Crank-Nicolson scheme
The idea in the Crank-Nicolson scheme is to apply centered differences in space and time, combined with an average in time. We demand the PDE to be fulfilled at the spatial mesh points, but in between the points in the time mesh: The important feature of this time discretization scheme is that we can implement one formula and then generate a family of well-known and widely used schemes: ,0 = 0

Conducting Von Neumann stability analysis on the Crank-Nicolson scheme.
In numerical algorithms for differential equations, the concern is the growth of round-off errors and/or initially small fluctuations in initial data, which might cause a large deviation of final answers from the exact solution. The method of stability analysis shown next was Von Neumann stability analysis.
To conduct the Von Neumann stability analysis on the Crank-Nicolson scheme, we need to apply the Fourier transform to the numerical solution. The Fourier transform converts the solution from the physical domain to the frequency domain, which allows us to analyze the stability of the scheme for different wave modes [1][2][3][4][5][6][7][8][9][10][11][12].
The scheme is stable if the amplification factor is less than or equal to 1 for all wave modes. The stability criterion can be determined by solving for the roots of the amplification factor equation in terms of and . The code created in Matlab calculates the Fourier coefficients for the initial state, defines the Fourier modes and calculates the amplification factor using the result obtained from the Crank-Nicolson scheme. It plots the region of stability in the − plane using the contour function. The stability region should lie within the unit circle, indicating that the circuit is stable for all wave modes.

Conclusion
In this work, we have analyzed the stability of the Crank-Nicolson scheme for the twodimensional diffusion equation using Von Neumann stability analysis. We have shown that the scheme is unconditionally stable, which means that it can be used for a wide range of parameters without being affected by numerical instability. We have also shown how to perform Von Neumann stability analysis on the scheme and plot the stability region in the Fourier space.
We have implemented the scheme in MATLAB and demonstrated its stability for a specific example equation. This approach can be generalized to other partial differential equations and used to design and analyze numerical methods for their solutions.
To solve the IBVP, we can start with the initial condition and use the finite difference equation to iteratively compute the solution at subsequent time levels . This can be done using various numerical methods, such as Gaussian elimination, LU decomposition, or iterative methods like Jacobi or Gauss-Seidel. We can solve the problem in the MATLAB program. In our next work, we show the stability of the Crank-Nicolson differential scheme for the two-dimensional diffusion equation.
In conclusion, the Crank-Nicolson scheme is a widely used numerical method for solving partial differential equations, and its stability is an important factor to consider when choosing a numerical method for a given problem. By understanding the stability properties of the Crank-Nicolson scheme, we can ensure the accuracy and reliability of numerical solutions of partial differential equations.