Question

The method of successive approximations (see Section \(2.8)\) can also be applied to systems of equations. For example, consider the initial value problem $$ \mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}, \quad \mathbf{x}(0)=\mathbf{x}^{0} $$ where \(\mathbf{A}\) is a constant matrix and \(\mathbf{x}^{0}\) a prescribed vector. (a) Assuming that a solution \(\mathbf{x}=\Phi(t)\) exists, show that it must satisfy the integral equation $$ \Phi(t)=\mathbf{x}^{0}+\int_{0}^{t} \mathbf{A} \phi(s) d s $$ (b) Start with the initial approximation \(\Phi^{(0)}(t)=\mathbf{x}^{0} .\) Substitute this expression for \(\Phi(s)\) in the right side of Eq. (ii) and obtain a new approximation \(\Phi^{(1)}(t) .\) Show that $$ \phi^{(1)}(t)=(1+\mathbf{A} t) \mathbf{x}^{0} $$ (c) Reppeat this process and thereby obtain a sequence of approximations \(\phi^{(0)}, \phi^{(1)}\), \(\phi^{(2)}, \ldots, \phi^{(n)}, \ldots\) Use an inductive argument to show that $$ \phi^{(n)}(t)=\left(1+A t+A^{2} \frac{2}{2 !}+\cdots+A^{x} \frac{r^{2}}{n !}\right) x^{0} $$ (d) Let \(n \rightarrow \infty\) and show that the solution of the initial value problem (i) is $$ \phi(t)=\exp (\mathbf{A} t) \mathbf{x}^{0} $$

Short answer

In this exercise, we used the method of successive approximations to find a general solution to the given initial value problem. We first derived the integral equation for the system and then calculated the first approximation for the solution. Through induction, we derived a general expression for the sequence of approximations and showed that it converges to the exponential of the matrix A multiplied by the scalar t and the initial vector x^0 as n approaches infinity. This is the general solution to the initial value problem for the given system of equations.

Step-by-step solution

Part (a): Derive the integral equation

From the given initial value problem: $$ \mathbf{x'}=\mathbf{A} \mathbf{x}, \quad \mathbf{x}(0)=\mathbf{x}^{0} $$ Let's assume that a solution \(\mathbf{x}=\Phi(t)\) exists. Then: $$ \Phi'(t)=\mathbf{A}\Phi(t) $$ Integrate this equation with respect to \(t\) from \(0\) to \(t\): $$ \int_{0}^{t}\Phi'(s)ds=\int_{0}^{t}\mathbf{A}\Phi(s)ds $$ This gives the integral equation for \(\Phi(t)\): $$ \Phi(t) - \Phi(0)=\int_{0}^{t}\mathbf{A}\Phi(s)ds $$ With the initial condition \(\Phi(0)=\mathbf{x}^0\), we can write the integral equation as: $$ \Phi(t)=\mathbf{x}^{0}+\int_{0}^{t}\mathbf{A}\Phi(s)ds $$

Part (b): First approximation\(\Phi^{(1)}(t)\)

Let's start with the initial approximation \(\Phi^{(0)}(t)=\mathbf{x}^{0}\). Substitute this into the integral equation: $$ \Phi^{(1)}(t)=\mathbf{x}^{0}+\int_{0}^{t}\mathbf{A}\Phi^{(0)}(s)ds $$ Since \(\Phi^{(0)}(t)=\mathbf{x}^{0}\), $$ \Phi^{(1)}(t)=\mathbf{x}^{0}+\int_{0}^{t}\mathbf{A}\mathbf{x}^{0}ds $$ Integrate with respect to \(s\): $$ \Phi^{(1)}(t)=\mathbf{x}^{0}+\mathbf{A}\mathbf{x}^{0}\int_{0}^{t}ds=\mathbf{x}^{0}+\mathbf{A}\mathbf{x}^{0}t $$ So, the first approximation \(\Phi^{(1)}(t)=(1+\mathbf{A}t)\mathbf{x}^{0}\).

Part (c): Calculate higher approximations

To derive a general expression for the sequence of approximations \(\Phi^{(n)}(t)\), we can apply an inductive argument: Assume that the following approximation is valid for some \(n\): $$ \Phi^{(n)}(t)=\left(I+\mathbf{A}t+\mathbf{A}^2\frac{t^2}{2!}+...+\mathbf{A}^n\frac{t^n}{n!}\right)\mathbf{x}^{0} $$ Now, let's find \(\Phi^{(n+1)}(t)\): $$ \Phi^{(n+1)}(t)=\mathbf{x}^{0}+\int_{0}^{t}\mathbf{A}\Phi^{(n)}(s)ds $$ Substitute the assumed expression for \(\Phi^{(n)}(s)\): $$ \Phi^{(n+1)}(t)=\mathbf{x}^{0}+\int_{0}^{t}\mathbf{A}\left(I+\mathbf{A}s+\mathbf{A}^2\frac{s^2}{2!}+...+\mathbf{A}^n\frac{s^n}{n!}\right)\mathbf{x}^{0}ds $$ Combine the integration with the sum and apply term-by-term integration: $$ \Phi^{(n+1)}(t)=\mathbf{x}^{0}+\left(\int_{0}^{t}\mathbf{A}ds+\int_{0}^{t}\mathbf{A}^2sds+\int_{0}^{t}\mathbf{A}^3\frac{s^2}{2!}ds+...+\int_{0}^{t}\mathbf{A}^{n+1}\frac{s^n}{n!}ds\right)\mathbf{x}^{0} $$ Calculate the integrals: $$ \Phi^{(n+1)}(t)=\mathbf{x}^{0}+\left(\mathbf{A}t+\mathbf{A}^2\frac{t^2}{2!}+\mathbf{A}^3\frac{t^3}{3!}+...+\mathbf{A}^{n+1}\frac{t^{n+1}}{(n+1)!}\right)\mathbf{x}^{0} $$ This proves that the assumed expression is valid for any \(n\).

Part (d): Convergence to the solution

Now, let's consider what happens when \(n \rightarrow \infty\). The expression for the sequence of approximations becomes: $$ \Phi(t)=\lim_{n\to\infty}\left(I+\mathbf{A}t+\mathbf{A}^2\frac{t^2}{2!}+...+\mathbf{A}^n\frac{t^n}{n!}\right)\mathbf{x}^{0} $$ This is the definition of the exponential of the matrix \(\mathbf{A}\) multiplied by the scalar \(t\): $$ \Phi(t)=\exp(\mathbf{A}t)\mathbf{x}^{0} $$ This is the solution to the initial value problem for the given system of equations.

Key concepts

Successive Approximations

Successive approximations represent a powerful method for solving initial value problems involving differential equations. The main idea behind this technique is to construct a sequence of approximate solutions that converges to the true solution. Starting with an initial guess, often the initial condition of the problem, you generate the next approximation by using the previous one in an integral or differential equation. The process resembles a feedback loop, where each iteration builds upon the previous one to improve accuracy.

In the context of solving the system of equations with a constant matrix, the first approximation is made by considering the initial condition as the entire solution. Each subsequent approximation is obtained by incorporating the effect of the constant matrix, representing the system's dynamic behaviour. Over time, if these approximations converge, they can provide a sufficiently accurate estimate of the system's true behaviour at different points in time.

To enhance comprehension, consider an analogy with photography: taking a picture is akin to capturing an initial condition, while successive approximations are like applying filters one by one, each layer refining the final image until it accurately represents the desired scene.

Constant Matrix

A constant matrix is simply a matrix with fixed numerical entries. In the context of differential equations, a constant matrix plays a central role when dealing with linear systems where the rate of change of the state vector \( \mathbf{x} \) is directly proportional to the current state and is governed by a set of linear relations. These systems are represented in the form \( \mathbf{x'}=\mathbf{A}\mathbf{x} \) where \( \mathbf{A} \) is the constant matrix. Its values dictate how different components of the state vector affect each other over time.

In our problem, the constant matrix affects the successive approximations because it scales the vector at each iteration. Understanding matrix operations and their implications on transformations is essential for grasping the problem's solution. Think of a constant matrix as a transformation tool that, when applied to a vector, skews, rotates, or scales its components, shaping the trajectory of the solution over time.

Integral Equation

An integral equation is a mathematical equation in which an unknown function appears under an integral sign. Unlike ordinary differential equations that relate the function and its derivatives, integral equations relate the function to its integrals. Integral equations are particularly useful when solving problems where a whole history (from the past to the present) influences the current state.

In our example, the integral equation arises from the accumulated effect of the constant matrix \( \mathbf{A} \) on the initial state vector \( \mathbf{x}^{0} \) over time. It encapsulates the entire evolution of the state vector from time zero to time \( t \) in a single expression. This integral equation forms the basis for the successive approximations, where at each step, the integral of the previous approximation's product with the constant matrix \( \mathbf{A} \) gives the next level of refinement to the solution.

To help visualize this, imagine filling a bathtub with water. The total water level (analogous to the integral of the solution) does not just depend on the current flow rate of water but also on all the water that has been added up to that moment. Similarly, the integral equation takes into account all the influences up to time \( t \) to determine the state of the system at that particular instant.