Application of Gauss Divergence theorem for linear Partial Differential Equation in the frame work of Green's Function

Abstract: a feasible region in three dimensions is enclosed by a surface [6] 𝑧 = 𝑓(𝑥,𝑦) whose dynamic equation 𝑥 = 𝑥(𝑡),𝑦 = 𝑦(𝑡),𝑧 = 𝑓(𝑥(𝑡),𝑦(𝑡))can be interpreted as the Gaussian surface found using the Gauss – Divergence theorem[4] that says “the outward drawn normal component multiplied with the tangent plane or the closed region as the projection of 𝑧 = 𝑓(𝑥,𝑦) or simply outward flux of the vector field κ upon the closed surface Σ is equal to the volume of the divergence of κ enclosed by Σ. Further, the flux using the arbitrary point that can be extended to the entire surface Σ which is equal to the circulation along the closed curve 𝔍 that encloses the projection 𝔑 of 𝑧 = 𝑓(𝑥,𝑦). Simply, a linear programming problem can be created from volume to the surface and then from the projection surface to the circulation. On the other hand, for the numerical data having more data points, for sake of ease, the method of cubic spline interpolation is considered optimal. A polynomial of higher degree can be written as the product of linear and quadratic polynomials in a complex plane by fundamental theorem of algebra. So, a linear 2nd order partial differential equation can be seen as two 2nd order ordinary differential equations by fixing each argument of one variable as constant and solve the equation for the 2nd variable. So, to solve a 2nd order differential equation that has two implicit variables t and s, one can apply a Green’s function and solve it. The action is suitable to both the explicit variables x and y, equating the solutions of both the differential equations at each case, we can get the generalization that represents the required integral surface. Key words: 2nd order linear partial differential operator, Outward flux, volume integral, Gauss Divergence theorem, boundary conditions, numerical solutions, enclosing curve, normal, surface integral, linearly independent solutions