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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]

In this paper, the element-free Galerkin (EFG) meshless method and moving Kriging collocation meshless technique are applied for finding the numerical solution of a class of two-dimensional (2D) nonlinear time fractional partial differential equations.The Klein–Gordon, sine-Gordon, diffusion wave and Cattaneo equations with Neumann boundary conditionare studied. The time fractional derivative has been described in the Caputo’s sense. Firstly, we use a semi-implicit finite difference scheme of convergence order, and then for obtaining a full discrete scheme, the space derivative is discretized with the EFG and moving Kriging collocation techniques. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method, test and trial functions are moving least squares approximation (MLS) shape functions. Also, in the element-free Galekin method, we do not use any triangular, quadrangular, or other types of meshes. The EFG method is a global method while finite element method is a local one. The EFG method is not a truly meshless method and for integration uses a background mesh. We prove the unconditional stability and obtain an error bound for the EFG method using the energy method. Numerical examples are reported which support the theoretical results and the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this research paper, we introduce a numerical approach to solve a particular type of nonlinear integro-differential equations derived from Volterra's population model. This model characterizes the growth of a biological species in a closed system and includes an integral term to consider the influence of toxin accumulation on the species, along with the conventional terms found in the logistic equation. The proposed technique estimates the solution of integro-differential equations utilizing the discrete Galerkin scheme using the moving least squares (MLS) algorithm. The locally weighted least squares polynomial fitting, known as the MLS method, is a valuable approach for approximating unknown functions. Since the offered scheme does not require any cell structures, it can be known as a meshless local discrete Galerkin method. Moreover, we obtain the error estimate of the proposed approach. The validity and efficiency of the newly developed technique are assessed over several nonlinear integro-differential equations. [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]