Obligation to Fast on the Yahrzeit

Why is it customary to fast on the yahrzeit, and is it obligatory?

There is a mitzvah to fast on the yahrzeit, and one must accept this fast upon oneself. If one fasted the first time and did not declare the fast as non-binding, one is prohibited from stopping the fast due to a vow. However, the Raak"a wrote that in our times, it is more appropriate not to fast due to the neglect of Torah study and to redeem the fast with charity, increased Torah study, and merits.
Regarding the reason for the fast: some have written that it is to save the deceased from the judgment of Gehenna (as the deceased is judged every day they passed away). Others have written that it is to save the son, as on the day of his father's death, his fortune is weakened.

It is stated in the Gemara, Shevuot, page 20a: "One who says: 'I am obligated not to eat meat and not to drink wine on the day my father died, on the day so-and-so died, on the day Gedaliah ben Ahikam was killed.'" From this, the Vilna Gaon in Siman 376, Subparagraph 7, and the Terumat HaDeshen, Siman 293, proved that one must fast on the day of a father's death. However, the Vilna Gaon says that some say this applies only to the actual day of death and not to all subsequent years on the date of death.
The Beit Yosef in Yoreh De'ah, Siman 373, did not bring the Gemara as a source for this law but brought in the name of the Tashbetz that one must fast on the day a father or mother died. This is also ruled by the Shulchan Aruch in Orach Chaim, Siman 548, Paragraph 8, and the Rema in Siman 572, Paragraph 12.
The Rema wrote: "However, if one completed it the first time, one should continue this practice all his life." The Pitchei Teshuva, Siman 572, Subparagraph 8, wrote: "See the commentary of Rabbi Maharit, and what he wrote, see Taz there." The Taz, Subparagraph 3, disagrees and wrote that it is not considered a vow unless one intends to practice this all his life, therefore one who is lenient does not lose, especially if he suffers. See the Pri Megadim in the commentary there, who wrote to be lenient if the first time (after 12 months) falls on Erev Shabbat. His opinion is that even if it occurs on a weekday, one should not complete it, as in our times, the generations are weak.
The reasons were brought by the Biur Halacha in the treatise Maamar HaKadishin, Siman 132, regarding Kaddish.
The Aruch HaShulchan, Yoreh De'ah, Siman 376, Paragraph 13, wrote: "And it is a mitzvah to fast on the day a father or mother died, and one who finds fasting difficult should redeem it with money and distribute it to the poor."