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The two-grid method is a technique to solve the linear system of algebraic equations for reducing the computational cost. In this study, the two-grid procedure has been combined with the EFG method for solving nonlinear partial differential equations. The two-grid FEM has been introduced in various forms. The well-known two-grid FEM is a three-step method that has been proposed by Bajpai and Nataraj (Comput. Math. Appl. 2014;68:2277-2291) that the new proposed scheme is an ecient procedure for solving important nonlinear partial differential equations such as Navier-Stokes equation. By applying shape functions of IMLS approximation in the EFG method, a new technique that is called interpolating EFG (IEFG) can be obtained. In the current investigation, we combine the two-grid algorithm with the IEFG method for solving the nonlinear Rosenau-regularized long-wave (RRLW) equation. In other hand, we demonstrate that solutions of steps 1, 2, and 3 exist and are unique and also we achieve an error estimate for them. Moreover, three test problems in one- and two-dimensional cases are given which support accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

Three inverse problems of reconstructing the time-dependent, spacewise- dependent and both initial condition and spacewise-dependent heat source in the one-dimensional heat equation are considered. These problems are reformulated by eliminating the unknown functions using some special assumptions concerning the points in space or time as additional measurements. Then direct techniques are proposed to solve the non-classical boundary value problems. For obtaining the robust and stable approximations, Bernstein multi-scaling and B-spline basis functions in the context of the Ritz–Galerkin method are utilized to immediate passage from differential equations to algebraic equations and afterwards, a Newton-type method is used to produce the admissible solution. The numerical convergence and stability are discussed in the test examples to show that the presented schemes provide accurate and acceptable approximations. [ABSTRACT FROM AUTHOR]

Three inverse problems of reconstructing the time-dependent, spacewise- dependent and both initial condition and spacewise-dependent heat source in the one-dimensional heat equation are considered. These problems are reformulated by eliminating the unknown functions using some special assumptions concerning the points in space or time as additional measurements. Then direct techniques are proposed to solve the non-classical boundary value problems. For obtaining the robust and stable approximations, Bernstein multi-scaling and B-spline basis functions in the context of the Ritz–Galerkin method are utilized to immediate passage from differential equations to algebraic equations and afterwards, a Newton-type method is used to produce the admissible solution. The numerical convergence and stability are discussed in the test examples to show that the presented schemes provide accurate and acceptable approximations. [ABSTRACT FROM AUTHOR]