# System of ODE Boundary Value Problem with 2nd Order Backward Difference

• teknodude
In summary, the conversation discusses a numerical problem in Matlab involving a second order backward difference formula. The issue is with the k-2 indice being outside the boundary and the suggestion is to use another numerical method, such as RK4, to start the process.
teknodude
$${\frac {{\it du}}{{\it dx}}}=998\,u+1998\,v$$
$${\frac {{\it dv}}{{\it dx}}}=-999\,u-1999\,v$$
$$u \left( 0 \right) =1$$
$$v \left( 0 \right) =0$$
0<x<10
Second Order Backward Difference formula
$${\frac {f_{{k-2}}-4\,f_{{k-1}}+3f_{{k}}}{h}}$$

I'm trying solve this numerically in matlab, but can't seem to figure out what to do with the k-2 indice in the 2nd order backward difference equation, because it is outside the boundary. I was thinking of using a ficticious or ghost point, but I thought that only applies if a neuman boundary condition is given. The way i think of it, I have 4 unknowns and only 2 equations.

EDIT: ok after thinking about it. I think I have to use another numerical method to start the process, like RK4.

Last edited:
I think that's a good idea xD

## 1. What is a System of ODE Boundary Value Problem with 2nd Order Backward Difference?

A system of ODE boundary value problem with 2nd order backward difference is a mathematical model that describes the behavior of a system of ordinary differential equations (ODEs) subject to boundary conditions. The 2nd order backward difference refers to the method used to numerically solve the system of ODEs, which involves approximating the derivatives of the functions using backward difference formulas.

## 2. What is the difference between a system of ODE Boundary Value Problem and a system of ODE Initial Value Problem?

A system of ODE boundary value problem is a type of problem where the values of the functions are known at the boundaries of the domain, while a system of ODE initial value problem is a type of problem where the values of the functions are known at a single point within the domain. In other words, a system of ODE boundary value problem involves solving for the functions at multiple points, while a system of ODE initial value problem involves solving for the functions at a single point.

## 3. What is the importance of using a 2nd Order Backward Difference in solving a System of ODE Boundary Value Problem?

The 2nd order backward difference method is commonly used in solving a system of ODE boundary value problem because it is a stable and accurate numerical method. It also takes into account the values of the functions at multiple points, making it suitable for solving boundary value problems. Additionally, it can handle stiff systems of ODEs, where the solution changes rapidly, without sacrificing accuracy.

## 4. How does one solve a System of ODE Boundary Value Problem with 2nd Order Backward Difference?

To solve a system of ODE boundary value problem with 2nd order backward difference, one must first discretize the problem by dividing the domain into a finite number of points. Then, the backward difference formulas are used to approximate the derivatives of the functions at each point. Finally, the resulting system of equations is solved using numerical methods, such as the Newton-Raphson method or the shooting method.

## 5. What are some applications of solving a System of ODE Boundary Value Problem with 2nd Order Backward Difference?

Solving a system of ODE boundary value problem with 2nd order backward difference has many applications in various fields such as engineering, physics, and economics. It can be used to model and analyze the behavior of physical systems, such as heat transfer and fluid flow. It can also be used in economic models to predict the behavior of financial markets. Additionally, it is a valuable tool in designing and optimizing engineering systems, such as control systems and structural designs.

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