3.2. Taylor Series Formulation

There is something called the Taylor Series Formulation which we generally use, usually finite difference equation consists of approximating the derivatives in the differential equations via a truncated Taylor series. So, how do we approximate the derivatives in the differential equation using a truncated Taylor series, which looks like this? I am pretty sure you have read that in your math class.

The Taylor Series formulation is a mathematical tool used to approximate functions through their derivatives around a specific point. In the context of partial differential equations (PDEs), we use a truncated Taylor series to express the derivative of a function at one point based on its values at nearby points. This makes it easier to compute approximate solutions for differential equations by turning them into finite difference equations.

Imagine you are trying to predict how far a ball will roll down a slope using the height of the slope. You can approximate this by looking at how the ball rolls from one point to another, much like how the Taylor series lets us predict the value of a function at one point by knowing its values at nearby points.

So, phi 1 is written as phi 2 - delta x del phi by del x at 2 + half delta x whole squared del squared phi into del x whole squared at 2. So, at series like this goes on with alternate - and + signs.

When we consider points in a series with the Taylor expansion, we express the function at point phi 1 in terms of its value at phi 2 and the derivatives of phi. The terms include not just the first derivative but also the second derivative, multiplied by the square of the step size (delta x) to account for how steep the curve is. This allows for a more accurate approximation of the function near phi 2. The alternating signs (+ and -) reflect the behavior of the derivatives as we consider them at various points.

Think of it like building a ramp. The first section may be steep (first derivative), but then it levels off (second derivative). The way you describe the ramp’s slope and angle will change depending on where you are measuring, just like the terms in the Taylor series try to capture the changing behavior of the function around the point.

Truncating the series just after the third term adding and subtracting the 2 equations so, you see there were 2 equations phi 1 and phi 3 both were return in terms of phi 2. If what we do if we just do you know if we first stopped the series just after the third term and add and subtract the 2 equation.

In this step, we simplify the calculations by stopping our Taylor series expansion after the third term. By adding and subtracting the equations for phi 1 and phi 3 (both expressed in terms of phi 2), we can isolate first and second derivatives of the function. This manipulation helps us form relationships that can be rewritten as finite difference equations, which are essential for numerical simulations of PDEs.

Consider a puzzle where you are trying to find the middle piece by only using pieces adjacent to each other. If you focus on three connecting pieces, you can deduce the shape and location of the middle piece without needing to see the entire board. This is similar to how we limit our series to three terms to clarify the relationships within the function.

The substitution of such expression into differential equation leads to finite difference equation.

Once we have these finite difference equations derived from our truncated Taylor series, we can substitute them back into the original differential equations. This conversion allows us to use numerical methods to approximate solutions rather than relying solely on analytical solutions that may be complex or impossible to derive. Essentially, finite differences provide a way to tackle differential equations by breaking them down into manageable calculations that can be handled numerically.

Imagine a chef trying to create a recipe from scratch. Instead of guessing all the ingredients at once (like solving a complex equation), they start with basic components (the finite differences) and gradually refine their dish, allowing for adjustments and improvements with each iteration based on taste tests (simulated calculations).

Analytical solution of the partial differential equation provide us with closed form expressions which depicts the variation of the independent variable in the domain, whereas, using the numerical simulation based on finite differences provide us with the values at discrete points in domain which are known at grid points.

The key difference between analytical and numerical solutions is that analytical solutions give us exact formulas that can predict values across an entire domain, while numerical solutions yield specific values at discrete points, making them suitable for more complex problems. Analytical solutions can be highly desirable for their precision, but many real-world problems require numerical methods due to their complexity.

Think of a map versus GPS navigation. A map provides you with all possible routes (an analytical solution), while a GPS will give you step-by-step directions for your journey based on your starting point (numerical solution). Both are useful, but they serve different needs depending on the situation.

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