@Mariusz Iwaniuk , Thank your for your time and efforts. It would be nice if you could tell me the way to calculate in mathematica. only if the forum rules allow.

@Rouben Rostamian  thank you for your reply. Its Newton's divided difference interpolation. However It helped me a lot to make my code for forward interpolation. For forward and backward interpolations the data has to be equispaced.

Thank you for sparing your time. Is there any other way in maple which can be used to achieve what I want?

I think something is missing. xL, xU, yL, yU should be known I think. Being them unknown is parhaps making the trouble.

every time i run this code the results are differnt and not as good as they seem in the graph above. In my view the values of TM are not stable and they change everytime i execute the code. I am using Maple 2017. Can you please explain what i am missing here.

@Kitonum @acer
thank you very much for your time. acer's answers was selected and a vote-up for both of you. Really appriciate your support.

Seems a very good book. Congratulations

I tired to use Expression tamplet to define the function that too failed. But unpply did the job which converts the expression to a function. Thank you @ThU . Thanks mapleprimes

@ThU I am really thankful to you for your reply. The chapter you refered is very old and the function definition was changed in later versions of maple (i am not exactly sure the exact version though). I can calculate f(2) or f(3) without any problem. The only problem that i face is f(0). I tried to define the function in the older way but still it gives the same output.
Download bug.mw

Aww that was a silly mistake. Thank you for pointing it out. Any suggestions to make the output bit more organized and elegant? For example if i want to display the resutls in a form of a table which may have the following coulumns
i    x[i]   y[i]    J[i]   f[1][i]    f2[i]

How to achieve that?

Can you please explain more what you want? do you want to change the values or swap them?

Thank you everyone for your kind help. one vote up for everyone.

I am sorry i was away . here is the manual method. Here_is_to_get_the_result_manually.pdf

@Carl Love here is the pdf raynor1982.pdf

@tomleslie  even if we take bet=0 there is still the same error. beta is MHD term so we can take it to be zero just to explore the remaining solution.