In Backward Euler, the equation to be solved at each step is
$$x_{n+1}=x_n+(t_{n+1}-t_n)f(x_{n+1},t_{n+1}).$$
That is, you want to find roots of the function
$$g_n(x)=x-x_n-(t_{n+1}-t_n)f(x,t_{n+1})$$
where we interpret $x_n,t_{n+1},t_n$ as parameters. So in particular the Jacobian for Newton's method is $I-(t_{n+1}-t_n)D_x f(x,t_{n+1})$, where $D_x$ represents differentiation with respect to the $x$ variables.
Consider a first order linear differential equation $y'+Py=Q,\,y(a)=y_0$ .Here continuity of $P$ and $Q$ ensure that the the ODE has an unique solution.But in case of non-linear initial value problem i.e $y'=f(x,y),\,y(x_0)=y_0$,continuity of $f$ does not ensure the unique solution.So we need to address the following question:
(1)Under what condition on $f$ the problem $y'=f(x,y),\,y(x_0)=y_0$ has a solution$?$
(2)If solution exists,whether it is unique or not$?$
$\to$First question is answered by the Peano existence theorem which states that "let $f$ be a continuous function in an interval $I$ containing the points $(x_0,y_0)$,then the the problem $y'=f(x,y),\,y(x_0)=y_0$ has a solution".
$\to$Second question is answered by Picard's uniqueness theorem which states that "let $f$ and $\frac{\delta f}{\delta y}$ are continuous in aregion R containing the initial points $(x_0,y_0)$ then the the problem $y'=f(x,y),\,y(x_0)=y_0$ has an unique solution"
Picard method for interval of definition:let $f$ and $\frac{\delta f}{\delta y}$ are continuous in a closed rectangle $$R=\{(x,y):|x-x_0|\leq a,|y-y_0|\leq b\}$$.Then the IVP $y'=f(x,y),\,y(x_0)=y_0$ has an unique solution in the interval $|x-x_0|\leq h=min{(a,\frac{b}{l})}$ where $l=MAX_{(x,y)\in R}|f(x,y)|$
Hope this will help you!!!
Best Answer
Picard's method of solving a differential equation (initial value problems) is one of successive approximation methods; that is, it is an iterative method in which the numerical results become more and more accurate, the more times it is used. Sometimes it is very difficult to obtain the solution of a differential equation. In such cases, the approximate solution of given differential equation can be obtained.
Picard's iteration method was initially used to prove the existence of an initial value problem. Although this method is not of practical interest and it is rarely used for actual determination of a solution to the initial value problem (due to slow convergence and obstacles with performing explicit integrations), nevertheless, there are known some improvements in this procedure that make it practical. Picard's iteration method is important because it leads to an equivalent integral formulation useful in establishing many numerical methods. i.e.,
First order initial value problems of the form $$y'(x)=f(x,y(x))~,~~~~y(x_0)=y_0$$ can be rewritten as an integral equation $$y(x)=y_0~+~\int_{x_0}^x~f(t,y(t))~dt~.$$ This integral formulation can be used to construct a sequence of approximate solutions to the IVP. The basic idea is that given an initial guess of the approximate solution to the IVP, say$~\phi_0(x)~,$ an infinite sequence of functions, $~\{\phi_n(x)\}~$, is constructed according to the rule $$\phi_{n+1}(x)=y_0~+~\int_{x_0}^x~f(t,\phi_n(t))~dt~.$$ That is the $n^{\text{th}}$ approximation is inserted into the right hand side of the integral equation in place of the exact solution $~y(x)~$ and used to compute the $~(n+1)^{\text{st}}~$element of the sequence.
Also, Picard's iteration method helps in developing algorithmic thinking when the user implements it in a computer. It was the first method to solve analytically nonlinear differential equations. Working with Picard's iterations and its refinements helps everyone to develop computational skills.