A Polynomial Finite Difference Technique to Generalize Helmholtz Type Equations

The Helmholtz differential equation is often used for efficient and dynamic modeling of wave scattering realworld problems. The time harmonic wave scattering phenomena find applications in several scientific areas such as acoustic, electromagnetism, sensors, seismic, radar, and solar technology. Lately, it has been vital in modeling medical imaging problems; an emerging need of humans. Helmholtz type’s differential equation(s) also arises from a physical phenomenon, therefore, it is important to solve Helmholtz types of equation(s) for several purposes. Analytically, it is difficult, in some cases impossible, to find the solution of such equation(s). The numerical method can be used to find the solutions of such equations. Therefore, for the numerical method, the finite difference method is simple and is widely used as compared to other methods. The second order central finite difference is usually used in solving the differential equation(s), but sometimes in many scientific applications we do desire to have higher order approximations and that desire stems from two issues first the better accuracy. Suppose we have done computations of a problem with moderate size mesh, but if the error that we are getting is not acceptable for some unknown reasons, at the same time it is difficult to solve it with a further finer mesh that can happen at times and that is where we want a higher order approximation which gives us the same accuracy as a finer mesh but with fewer mesh points. This is the basic idea and memory storage is another important issue. The discretization of the space differential in the finite difference method is usually derived using the Taylor series expansion; however, if we use a method that adopts algebraic polynomial interpolations in the calculation around near-wall elements, all the calculations over irregular domains reduce to those over regular domains. However, if we use the polynomial interpolation systematically, exceptional advantages are gained in deriving high-order differences which is explained in detail in this paper with examples.