Abstract:
This research tackles solving a broad class of second-order partial differential
equations (PDEs) with a novel method using natural cubic splines. These equations, crucial in
science and engineering, describe phenomena like heat flow and wave propagation. The
method works for both parabolic (diffusion) and hyperbolic (wave) equations, with a focus
here on parabolic ones. The key idea lies in approximating the spatial derivatives in the PDE
with the second derivative of a natural cubic spline function. Imagine a smooth curve broken
into segments; natural cubic splines ensure these segments connect seamlessly while having
zero second derivative at the joints. This makes them ideal for mimicking the solution's
behavior. For the time derivatives, the paper employs a finite difference method. This
approximates the derivative based on the function's values at specific time steps. By combining
these approximations, the original PDE transforms into a system of solvable algebraic
equations. The paper explores solving this system explicitly (directly calculating new solutions
based on previous ones) and implicitly (solving a system of equations at each step). This offers
flexibility, with explicit schemes being faster but potentially less stable, while implicit schemes
provide more stability but require more computation. Finally, the paper validates the method's
effectiveness through numerical examples with various boundary conditions (specifying the
solution's behavior at domain edges). This showcases the method's applicability in real-world
scenarios with different constraints. In conclusion, this research offers a valuable tool for
solving diverse second-order PDEs. The method's ability to handle both constant and variable
coefficients and its exploration of different solution strategies make it a versatile and adaptable
approach.
