Kansa method

The Kansa method is a computer method used to solve partial differential equations. Partial differential equations are mathematical models of things like stresses in a car's body, air flow around a wing, the shock wave in front of a supersonic airplane, quantum mechanical model of an atom, ocean waves, socio-economic models, digital image processing etc. The computer takes the known quantities such as pressure, temperature, air velocity, stress, and then uses the laws of physics to figure out what the rest of the quantities should be like a puzzle being fit together. Then, for example, the stresses in various parts of a car can be determined when that car hits a bump at 70 miles per hour. The Kansa method is a computer method used to solve partial differential equations. Partial differential equations are mathematical models of things like stresses in a car's body, air flow around a wing, the shock wave in front of a supersonic airplane, quantum mechanical model of an atom, ocean waves, socio-economic models, digital image processing etc. The computer takes the known quantities such as pressure, temperature, air velocity, stress, and then uses the laws of physics to figure out what the rest of the quantities should be like a puzzle being fit together. Then, for example, the stresses in various parts of a car can be determined when that car hits a bump at 70 miles per hour. The Kansa Method can be explained by an analogy to a basketball court with many light bulbs suspended all across the ceiling. If the brightness of each bulb can be individually adjusted, any desired light intensity pattern at every x, y point on the floor of the basketball court can be approximated. This light intensity pattern on the floor of the basketball court is the approximate solution to a partial differential equation. The Kansa Method is mathematically much easier to understand and program than the finite element method. It works very well when the number of variables exceed x,y,z, and time. E. J. Kansa in very early 1990s made the first attempt to extend radial basis function (RBF), which was then quite popular in scattered data processing and function approximation, to the solution of partial differential equations in the strong-form collocation formulation. His RBF collocation approach is inherently meshless, easy-to-program, and mathematically very simple to learn. Before long, this method is known as the Kansa method in academic community. Because the RBF uses the one-dimensional Euclidean distance variable irrespective of dimensionality, the Kansa method is independent of dimensionality and geometric complexity of problems of interest. The method is a domain-type numerical technique in the sense that the problem is discretized not only on the boundary to satisfy boundary conditions but also inside domain to satisfy governing equation. In contrast, there is another type of RBF numerical methods, called boundary-type RBF collocation method, such as the method of fundamental solution, boundary knot method, singular boundary method, boundary particle method, and regularized meshless method, in which the basis functions, also known as kernel function, satisfy the governing equation and are often fundamental solution or general solution of governing equation. Consequently, only boundary discretization is required. Since the RBF in the Kansa method does not necessarily satisfy the governing equation, one has more freedom to choose a RBF. The most popular RBF in the Kansa method is the multiquadric (MQ), which usually shows spectral accuracy if an appropriate shape parameter is chosen. The Kansa method, also called modified MQ scheme or MQ collocation method, originated from the well-known MQ interpolation. The efficiency and applicability of this method have been verified in a wide range of problems. Compared with the boundary-type RBF collocation methods, the Kansa method has wider applicability to problems whose fundamental and general solutions are not available, e.g., varying coefficient and nonlinear problems. Let d-dimensional physical domain Ω ⊆ R d {displaystyle Omega subseteq mathbb {R} ^{d}} and consider the following boundary value problem (BVP)