I'm troubled by the preponderance of solutioning instead of designing. The OP said, "I want to know what is the most usable and problem free solution, so user wont be confused at all." So whether it's jQuery or JavaScript or C# or FORTRAN, the design is important--and it's the UX design, not the UI design that matters here. (Another plea to avoid the incorrect and illogical construct "UX/UI").
Use text input. Use three separate boxes labeled Day, Month, and Year (or Month, Day, and Year). Let users type the dates. Mixing text and lists in the same interaction is a Bad Thing (don't use text input for year but date picker or lists for month and day, for example).
Use a single text field that uses in-field format hints, for example, [ day / month / year ] Don't require formats too rigidly. If a user type JUN in the place where you expect a month, then use June on the back end. If a user types 6 in the place where you expect a month, then use June on the back end. If a user types 06 in the place where you expect a month, then use June on the back end.
Use Occam's Razor as a guide (in effect, most of the time the data will be accurate enough). Reduce cognitive friction (don't make users think).