B-Spline Solution of Partial Differential Equations. Applications of B-spline functions in the numerical solution of partial differential equations

      Аннотация: This work considers the numerical approximation of differential equations by using the B-spline method. The following types of problems in differential equations are investigated: • Second, third and fifth-order nonlinear boundary-value problems in partial differential equations. • Third-order nonlinear boundary value coupled equations in partial differential equations. Cubic, quartic, sextic B-splines are introduced. Some well-known results and a preliminary discussion about stability analysis of the boundary value problems was described. Cubic B-spline functions are used to develop numerical methods for computing approximations to the solution of second order nonlinear partial differential equations of parabolic and hyperbolic types. Also, numerical methods for computing approximations to the solution of nonlinear third-order boundary value problems are developed using quartic B-splines. Finally, a numerical method based on sextic B-spline method is used for solving nonlinear fifth-order boundary value problems.

Ключевые слова: B-spline functions, second order parabolic equations, fifth order partial equations, Newell Segel type equation and Phi-four, third order partial differential equations known as Gardner, Harry Dym and Jaulent-Miodek coupled equations