Modified Bessel Equations 홍성민

General Bessel Equation of order n: (1) The general solution of Eq.(1) Let’s consider the solutions of the 4 types of modified Bessel equation. Before we start, let’s remind the general form of the Bessel equation and solution. (2) General Bessel Equation

Let’s consider the following form of the Bessel Equation. (3) To solve this, first Let Then, (4) (5) Modified Bessel Equation 1

By substituting Eq.(4),(5) into (3), we are able to get the following equation. (6) (7) If we simply change x to t, we can catch that the Eq.(7) has a same form with the Eq.(1). By the Eq(2), the general solution of the Eq(1) is following. Since, The general solution is (8) Modified Bessel Equation 1

Now, let’s consider another type, the following Bessel Equation. (9) Similarly to the previous case, Let’s assume that. Then we get following two relations. (10) (11) Modified Bessel Equation 2

Plug in the Eq(10),(11) into Eq(9), then we can get the following equation. Now, let’s rearrange the terms of the above equation by the order of dz/dx. Modified Bessel Equation 2

Here, let’s divide both side of the equation on the previous slide by and rearrange the terms to make the equation mush simple. As a result, we can get the Eq.(12). (12) Modified Bessel Equation 2

Let’s assume again that Then, we can get the following relations by taking derivatives. Let’s substitute the two derivatives above into the Eq.(12). I will show you the detail on the next slide. Modified Bessel Equation 2

By substituting, we can finally rewrite the Eq.(12) to Eq.(13). Let’s consider the detail. First, by simply substituting we get this equation. Then, rearrange the terms by the order of dz/dt and simplify them! By dividing both side by, we can finally get the Eq.(13) (13) Modified Bessel Equation 2

Now, let’s plug t instead of then we get the Eq.(14) (14) This Eq.(14) is the case of the Modified Bessel Equation 1 assumed by Thus, we can say that the solution of the Eq.(14) is Therefore, the solution of the Eq.(8) is (15) Modified Bessel Equation 2 And we assumed that

Now, let’s consider the third type of the Modified Bessel Equation. This equation has the following form. (16) Here, let’s remind the first case of modified Bessel Equation, Eq.(3). (3) Compare the Eq.(16) and Eq.(3). We can easily catch that Eq.(16) has same form of Modified Bessel Equation 1 if we assume that. Since the Eq.(3) has the following solution, The solution of the Modified Bessel Equation 3 is (17) (18) Modified Bessel Equation 3

Let’s consider the last Modified Bessel Equation. It has the following form. (19) Multiplying to both side of the Eq.(19). (20) Let’s assume that, Then, we can get the following relations. (21) (22) Modified Bessel Equation 4

Substituting Eq.(21),(22) into Eq.(20). Rearrange by the order of dz/dx and simplify. Divide both side by x. (23) Modified Bessel Equation 4

Here, let’s assume that Then, we are able to get the following relations. (24) (25) Put the result of Eq.(24),(25) into (23). (26) Modified Bessel Equation 4

Eq.(26) is the same form of [Modified Bessel Equation 3], it has the following Solution. Here,, so the solution should be the following form. (27) Here, let’s denote I and K as following. (28) (29) Since n=1, we can rewrite the above terms by putting 1 instead of n. (30) (31)

Modified Bessel Equation 4 We can rearrange the Eq.(30) and (31). First, let’s rearrange Eq.(30) by the term of J. The result is Eq.(32). Next, rearrange Eq.(31) by the term of Y to put the results into the Eq.(27). The process is following. (32) (33) Now, by substituting the Eq.(32),(33) into the Eq.(27), we can get the following Result, the Eq.(34) (34)

Modified Bessel Equation 4 Here, let’s set the new constants C and D as following. Then, we are able to rewrite the Eq.(34) by changing the constants. We finally get the Eq.(35) (34) (35) As we assumed that, The solution of the Modified Bessel Equation 4 is