On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Abstract

Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second-order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one-dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article.

Solving nonlocal initial‐boundary value problems for parabolic and hyperbolic integro‐differential equations in reproducing kernel hilbert space

“…According to [27], the space W (3,1) is a reproducing kernel Hilbert space and similar to [28], we can obtain its reproducing kernel R (3,1) ( …”

“…Remark 2.2. We remark that, for every u(x, t) ∈ W (4,2) u( W (3,1) Now, we will give the representation of analytical solution to (6)- (9) in W (3,1) .…”

“…Now, suppose that ϑ i (x, t) ∞ i=1 be a normalized orthogonal system in W (3,1) , such that ϑ i (x, t) = i k=1 δ ik ϑ k (x, t), i = 1, 2, · · · , then we state the following theorem that gives the exact solution. Theorem 3.3 (see [26]).…”

“…We now study the convergence of proposed method under some hypotheses which provide the theoretical basis of the proposed method and further present the error estimates for obtained approximation in W (3,1) .…”

“…The study of boundary value problems with integral conditions has attracted many researchers due to its wide application in engineering in many areas, see, for example, [1][2][3].…”