Well, the reason for wanting a proposal is simple - to prepare one takes effort, and that effort is a good gauge of how committed a student is towards undertaking a PhD. Applying to become a PhD candidate can be narrowed down to the following steps:

Step 1: Selecting A Field Of Study
Since you already have a field of study selected, we'll jump over to step 2.

Step 2: Identifying A Supervisor
Google recent studies in the field to get a feel of what research is being done at the moment. From there, you'll be able to make a list of researchers (local/foreign) that are publishing meaningful work in the field. Look up their institutional/personal webpages, and read up on work they've done. If any of their work interests you, contact them to request a meeting, stating your interest in their work. Bring along a resume and your transcript so they can see what experience you have and how you fared in your previous courses. Each researcher has his/her target outcomes, and if you fit into their plans, they'll usually have a list of available topics for you to choose from.

Step 3: Topic Selection, Or More Commonly Known As The Case Of Proposing Your Own Topic VS Selecting A Topic As Proposed By Your Supervisor
This is a bit of a tricky one. Students are generally advised to go with supervisor-suggested topics for one reason - there's a lesser chance of you getting into a situation where you're stuck with your research and your supervisor is unable to assist you. If your target supervisor doesn't offer you a topic which you're comfortable with, walk away, there are plenty of other potential supervisors out there. Don't be cornered into doing something you hate, it'll only translate to poor results later on.

Step 4: Proposal Preparation
Preparing a research proposal isn't exactly rocket science, but many candidates have difficulty in preparing a decent one. My suggestion is that you ask your supervisor for a sample proposal that one of his/her previous students prepared and work off that. A good research proposal addresses the following concerns:
i. Research relevance - Is there a need for this research? If yes, why? What research has been done in the past that led you to this topic?
ii. Research viability - Can this research be completed on schedule? Are the target outcomes realistic? Does your selected institution have the physical requirements (lab equipment etc.) to make this research possible?
Remember that because you're preparing this proposal prior to starting your research, it doesn't have to be very detailed. However, you do need to do some reading up so you can prepare a meaningful proposal. Your supervisor should be able to guide you towards papers/books that you can consult prior to preparing your proposal. Last but not least, don't forget a Gantt chart!

Step 5: Formal Application
If you've been accepted by a researcher at a given institution as a student, the formal application is usually a breeze. In addition to your research proposal, a supporting letter from your supervisor of choice will help smooth out the application process. Since you're applying with scholarship, don't forget a letter for guarantee of payment from your sponsor.

I hope this helps mate, best of luck!

If you're looking at applied math, do look up both these researchers:

Reza Saatchi - Prof. Reza works mostly with electrical engineers, but he does require mathematicians to work on digital signal processing. Among some of his projects are medical equipment for non-invasive measurements.

Chrystopher Nehaniv - Prof. Nehaniv is mainly an algebraist, but he has worked with applied topology as well. One project which he consulted on caught my attention - using topology to map the surface of a fish's skin, allowing smart scanners to identify and segregate fishes caught into different groups (breed/quality). If academic lineage matters to you, Prof. Nehaniv was once supervised by John Rhodes, who is well known for founding Krohn-Rhodes theory.