s. s. dragomir

The mathematical description of various processes such as the nonlinear Klein-Gordon equation occurring in mathematical physics leads to a nonlinear partial differential equation. The mathematical model is only the first step, however, towards finding the solution of the problem under consideration. It has become possible to develop realistic mathematical models with the currently available computing power for complicated problems in science and engineering. To the best of our knowledge, systematically using the collocation method to acquire the numerical solution has not been previously used for the Klein-Gordon equation. The main aim of this paper is to systematically use the collocation method to acquire the numerical solution of the two coupled nonlinear non-homogeneous Klein-Gordon partial differential equations. We examine and analyze their stability, in detail. To this aim, we use the Von Neumann stability method to show that the proposed method is conditionally stable. A numerical example is introduced to demonstrate the performance and the efficiency of the proposed method for solving the coupled nonlinear non-homogeneous Klein-Gordon partial differential equations. The numerical results demonstrated that the proposed algorithm is efficient, accurate, and compares favorably with the analytical solutions.

‎Let $left( H;leftlangle cdot‎ ,‎cdot rightrangle right)$ be a complex‎ ‎Hilbert space‎. ‎Denote by $mathcal{B}left( Hright)$ the Banach $C^{ast }$-‎algebra of bounded linear operators on $H$‎. ‎For $Ain mathcal{B}left(‎Hright)$ we define the modulus of $A$ by $leftvert Arightvert‎ :‎=left(‎A^{ast }Aright) ^{1/2}$ and $func{Re}A:=frac{1}{2}left( A^{ast‎‎}+Aright)‎.‎$ In this paper we show among other that‎, ‎if $A,$ $Bin mathcal{‎‎B}left( Hright)$ with $0leq mleq leftvert left( 1-tright)‎‎A+tBrightvert ^{2}leq M$ for all $tin left[ 0,1right]‎,‎$ then begin{align*}‎ ‎0& leq int_{0}^{1}fleft( leftvert left( 1-tright) A+tBrightvert‎‎^{2}right) dt-fleft( frac{leftvert Arightvert ^{2}+func{Re}left(‎‎B^{ast }Aright)‎ +‎leftvert Brightvert ^{2}}{3}right) \‎ ‎& leq 2left[ frac{fleft( mright)‎ +‎fleft( Mright) }{2}-fleft( frac{‎m+M}{2}right) right] 1_{H}‎ ‎end{align*} ‎for operator convex functions $f:[0,infty )rightarrow mathbb{R}$‎. ‎Applications for power and logarithmic functions are also provided‎.