Praise be to Allah, the Lord of the World; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
Allah Orders His slaves to supplicate Him in all their needs and He promises to respond to their supplication by his Mercy and Favor only. Allah Says (interpretation of meaning): {And when My slaves ask you (O Muhammad concerning Me, then (answer them), I am indeed near (to them by My Knowledge). I respond to the invocations of the supplicant when he calls on Me (without any mediator or intercessor). So let them obey Me and believe in Me, so that they may be led aright.}[2:186].
There are many specific times when Allah responds to supplications, thus a person should seek those times to supplicate Allah. Among them are the last one third of the night, the period between Azan and Iqamah, right after finishing the obligatory prayers and when the Imam enters the pulpit on Friday till the end of the prayer.
Also one should supplicate the Dua' of the Prophet Younus (Alaihi al-Salaam) when he was in the stomach of the fish. The Prophet said: "No Muslim will supplicate Allah for anything and fail to receive an answer when he uses Dhun-Nun's supplication when he was in the belly of the fish. "لاَ إِلَهَ إِلاَّ أَنْتَ سُبْحَانَكَ إِنِّي كُنْتُ مِنَ الظَّالِمِينَ." "There is no god but Thee to whom be the glory. I was indeed a wrongdoer" [Tirmizi].
A supplicating person should not be in hurry for a response to his Dua' and should not lose hope of a response to his Dua for the above Hadith.
The Prophet said: "The invocation of anyone of you is granted if he does not show impatience by saying, I invoked Allah but my request has not been accepted!" [Reported by Imams Bukhari and Muslim].
Therefore, you should make Dua' during the above mentioned times, not despair of an answer to the Dua', continue the Dua' of Dhun-Nun, and ask Allah with all submission and humility, surely Allah will accept your supplications.
Allah knows best.