On well‐posedness of nonclassical problems for elliptic equations

Abstract

In the present paper, we consider nonclassical problems for multidimensional elliptic equations. A finite difference method for solving these nonlocal boundary value problems is presented. Stability, almost coercive stability and coercive stability for the solutions of first and second orders of approximation are obtained. The theoretical statements for the solutions of these difference schemes are supported by numerical examples for the two‐dimensional elliptic equations. Copyright © 2014 John Wiley & Sons, Ltd.

Numerical Algorithm for the Third-Order Partial Differential Equation with Three-Point Boundary Value Problem

“…It is obvious that 4 equations can be obtained from boundary value conditions and (6) give 10 equations. According to (7), we have 12 equations. Finally, 12 equations follow from the continuity at .…”

“…Many physical phenomena and mechanical situations have been formulated into boundary value problems with integral boundary conditions [4,5]. Later many works have appeared such as Ashyralyev and Aggez [6], Ashyralyev and Tetikoglua [7], Pulkina [8], and Ashyralyev and Gercek [9]. It should be noted that there are so much work devoted to the existence of solution for this type of boundary value problems where parabolic equations, hyperbolic equations, and mixed-type equations are considered [10,11].…”

“…Motivated by all the works above, we describe an improvement of the RKSM to find the numerical solution for (1). A new RKS is successfully established by some techniques like (3) and (7). Furthermore, other partial differential equations with multipoint boundary value conditions may be numerically solved using a similar process.…”

Numerical solution of the nonlocal boundary value problem for elliptic equations

“…Therefore, mathematical models of various physical, chemical, biological or environmental processes often involve nonclassical conditions. The well-posedness of various nonlocal boundary value problems for partial differential and difference equations has been studied extensively by many researchers (see, e.g., [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and the references given therein).…”