Abstract
We investigate the effectiveness of reproducing kernel method (RKM) in solving partial differential equations. We propose a reproducing kernel method for solving the telegraph equation with initial and boundary conditions based on reproducing kernel theory. Its exact solution is represented in the form of a series in reproducing kernel Hilbert space. Some numerical examples are given in order to demonstrate the accuracy of this method. The results obtained from this method are compared with the exact solutions and other methods. Results of numerical examples show that this method is simple, effective, and easy to use.
Highlights
The hyperbolic partial differential equations model the vibrations of structures
The comparison between interpolating scaling function method [11] and RKM for different values of α, β, and t is given in Tables 4 and 5
Numerical solutions are described in the extended domain [−4, 4] × [−3, 3]
Summary
The hyperbolic partial differential equations model the vibrations of structures (e.g., buildings, beams, and machines). These equations are the basis for fundamental equations of atomic physics. We consider the telegraph equation of the form ∂2u ∂t2 (x, t) + ∂u ∂t β2u (1) = ∂2u ∂x2 f (x, t) ,. 0 ≤ x, t ≤ 1, α > β ≥ 0, with initial conditions u (x, 0) = φ1 (x) , ut (x, 0) = φ2 (x) , (2). Appropriate boundary conditions u (0, t) = g0 (t) , u (1, t) = g1 (t) , t ≥ 0 (3)
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