I’ve been learning the piano for a year now but I’m still having trouble developing the strength and independence of the fourth finger. Can anyone suggest and useful exercises that will help me in this?
You must go as slow as necessary for your 4th finger to "consciously" make a move. If you are not experiencing the specific instruction of the brain to move the 4th finger to a spot with the correct form, rhythm, articulation, and dynamic, you are going too fast. As the saying goes, the chain is only as strong as the weakest link, or as in hiking, the rule that the group should only go as fast as the slowest hiker. Playing any exercises too fast will not strengthen the weaker fingers but will cause dependence on the stronger fingers to do the work. The weaker fingers will then just be "dragged along" and will not develop their own independence. You must go slow, slow, slow. This will also help to prevent injury. It may seem for a while that no changes are taking place, but the gradual strengthening does increase significantly over time. Be patient with yourself.
There are standard exercises used by many piano players in order to improve the technique and strength. Do a search on imslp.org for Hanon or Czerny. (Imslp.org has freely downloadable scores that are now free to use). This link is one example: https://imslp.org/wiki/Practical_Exercises_for_Beginners%2C_Op.599_(Czerny%2C_Carl)
The third and fourth fingers naturally have a bit less independence from each other than the others. To see this for yourself, simply try to touch the base of each of your fingers with its tip (curl your finger as much as you can, in other words). Your thumb is your only fully independent finger.
"Finger independence" is therefore something of a misnomer. However, what you can do with your fingers is learn how to flex them partially without involving any of your other fingers. Try this: hold out your hand flat. Then bend your fingers individually at the first knuckle, the joint connecting the finger to the hand. See if you can bend it to 90 degrees, without it shaking and without moving the other fingers.
When you can do that, see if you can bend the finger at the second joint (the middle joint of the three in each finger other than the thumb) as far as it will go, without moving the other fingers. You should come pretty close to being able to tap your palm with your fourth finger.
You now have, in effect, two basic finger motions that you can use on the piano: dropping a finger flat onto the key, and a sort of plucking motion that's useful for staccato. You can experiment with these two and motions in between (combining both of them in different ways) and see what works for you musically and technically.
If you have been playing for only a year then I would suggest that it is too soon to consider that you have a problem. Do you have a teacher? If so what do they think?
Get yourself some good sets of studies - not specifically for 4th and 5th fingers, but general - and work with them. Improve your general abilities. Practice and repetition works wonders.
If you still have a problem in two years time (which I would doubt if you work hard) then perhaps you do need some more specialized help.
There are all sorts of studies. Do not go for Chopin, Liszt or Scriabin: they will be too difficult for you at this stage so you will get very little out of them. I quite like Czerny and there are loads of them. I do not like Hanon (mostly because they are so very boring). There is also Cramer which, in my view, are very good indeed.
No exercises, books or strength building gymnastics are necessary. The four and five fingers are just as strong and coordinate as all your others. Your fingers have no muscles and are controlled by the arm. So . . .
Spread out your fingers and notice that the two and three are pretty much aligned with your forearm. The weight of the arm gives them their power merely through alignment. Notice also that the four and five are not aligned with the arm and the fulcrum of the knuckle is broken. All you need to do is train yourself to realign the forearm each time you use the four and five so that the fingers are in a straight line with the forearm. CARE MUST BE TAKEN NOT TO DEVIATE RADIALLY. Know that it is the arm that places each of the fingers. The fingers don't drag the arm behind it. Think of playing the piano like washing a window, waxing a car or writing on a chalk board.
Other things that get in the way of the four and five giving them the illusion of weakness is abducting the fingers, that is what you just did. The five fingers operate most efficiently when they are kept together. Spread them out and you are now using flexors and abductors simultaneously which will then pull on one another making them feel weak and permitting tension to creep in.
The five fingers prefer to all play together. When you go from one to two, for instance, the three, four and five must play in the same direction. Obviously only the two presses a key down but the others must maintain the same direction. By isolating them you create force vectors where your bone is pulled in two directions. In normal playing, the direction of the arm is constantly changing. We develop fatigue, pain and uncoordinated movement when we isolate the fingers and try to play from them. I have no idea why the prevailing pedagogy endorses isolating fingers. It is not how we are built.
So, in addition to playing with gravity or, the weight of the arm, which eliminates the need for strength, you can employ your pronator and supinator. The supinator is very powerful and doesn't fatigue. If combined with up and down, gravity, alignment, rotation, in and out . . . all those movements done together will make your fingers feel effortless and make them all feel equally powerful.
Find a teacher who knows about physics and ergonomics. Ask them if they know what is ulnar deviation, the pronator, why not to cross the thumb under the palm, the importance of in/out, where the key is heaviest, when to abduct . . . if they don't know the answer to any of those questions, you have a teacher who could eventually cripple you. RUN.