2
votes
0answers
23 views

Solution of partial difference equation

I want to find the explicit solution of the following difference equation $e_{i,j+1}=re_{i-1,j}+(1-2r)e_{i,j}+re_{i+1,j}+km_{i,j}$ where $r>0$, $k>0$ and $m_{i,j}$ are known and $e_{i,0}=0$. ...
0
votes
1answer
46 views

Solving ODE with matrices

I have an equation in ODE $M{'}(x)= M(x)*A(x)$. Issue here is $A(x) = C_1+C_2* x $ where $C_1,C_2 $ has dimension $3 \times 3$. And x is a scalar variable Doubt What is M(x)? Can any one give ...
0
votes
0answers
61 views

Find the solution for the given equation in discrete finite differences

Given this equation, determine its solution in finite differences: $y(n+2)+y(n+1)+y(n)=0$ With initial conditions: $y(0)=0$, $y(1)=0$. To determine the general solution I got this: $\lambda^2 + ...
0
votes
0answers
75 views

Catalan numbers via partial difference equations?

It is known that Catalan numbers can be characterized in the following way: let $f(n,k)$ be a function of two integer variables, such that the following recurrence holds: $$f(n+1,k+1) = f(n,k) + ...
5
votes
3answers
661 views

Chain rule for discrete/finite calculus

In the context of discrete calculus, or calculus of finite differences, is there a theorem like the chain rule that can express the finite forward difference of a composition $∆(f\circ g)$ in ...
0
votes
1answer
99 views

Sum $\sum_{k=0}^n p(k) \cdot f(k)$ in terms of $f(n)$ and $\sum_{k=0}^n f(k)$

I am aware of that this question shall be rather basic, and that there may be a lot of resources on this, but it is quite complicated to use Google to find relevant results for this (I have not found ...
3
votes
2answers
2k views

Function as parameter in Wolfram Mathematica

I want to define some basic functions known from "discrete analysis": $$I(f)(x):=f(x)$$ $$E(f)(x):=f(x+1)$$ $$\Delta(f)(x) := (E-I)(f)(x) = f(x+1)-f(x)$$ $$\nabla(f)(x) := (I-E^{-1})(f)(x) = ...