Thank you all for your cooperation and help!Maybe I should have started by explaining the final goal. By making a vertical straight line containing part of the image and then stretching that line left or right, I want to get the example shown below.

In the example shown, this effect was created by making straight vertikal line copy of image and it looks good.

But it doesn't work so well with other images. For this reason, it is better when I make a copy of the end, (the edge) of the image. Thus, the stretch of the copy will look much better together with the image.
I will try with Tas_mania's 'The snake' solution and share the results as soon as I can.
Thank you, guys

P.S. As I looking for PhotoComix Smoothing from G'MIC I have found G'MIC Online (
https://gmicol.greyc.fr/) - a web service to allow users applying image processing algorithms on their images, directly from a web browser. There's a "Pixel Sort" plugin out there that did the job for me.