tokens: fourth equation of 11 ), to derive j x , 1, out . Then, if we take into account ( B6 ) and the third equation of ( B4 ), we get: where To obtain the first amendment N 1, out , we apply the fact that . Then, we take account ( B9 ), and N 1, out takes the following form: where and . N 1, ihs can be derived if we use the flow definition for the HS medium j x , 1, ihs that is proportional to and take into account ( B6 ). Then, we obtain: The first amendment N 1, ihs can be presented as follows: where . Here, the continuity condition of densities at HP, i.e. the condition N 0, ihs ( x hp , η) + N 1, ihs ( x hp , η) = N 0, out ( x hp , η) + N 1, out ( x hp , η) was applied (the second equation of 11 ). After substituting the proposed form in ( B12 ), we obtain: Due to the fact that the solution of the homogeneous equation ( B14 ) has the form and taking into account the condition where Φ( x = x hp , η) = 0, it is necessary to search for an Φ( x , η) in the following form: Consequently, by taking into account ( B15 ) and ( B14 ), Φ( x , η) takes the following form: where Note that S 2 ( x ), S 3 ( x ), and S 4 ( x ) were obtained in consequence of expanding of the exponent from ( B15 ) in a Taylor series and integrating appropriate expressions to derive Φ( x , η). Consequently, the first amendment N 1, ihs ( x , η) takes the following form: Now, we use, on the one hand, the flow definition for the inner region of the heliosphere j x , 1, in what is proportional to and, on the other hand, the value of its expression that is determined by ( B5 ). Then, we obtain Let us present the first amendment N 1, in in the following form: where , i.e. the continuity condition of densities at TS, namely N 0, in (1, η) + N 1, in (1, η) = N 0, ihs (1, η) + N 1, ihs (1, η) is applied (the first equation of 11 ). After substituting ( B19 ) in ( B18 ), we obtain The solution of the homogeneous equation ( B20 ) has the form , this means that a solution for Υ( x , η) should be sought in the following form: As a result, by taking into account the form ( B21 ) and the expression ( B20 ), we get Υ( x , η). After substituting it in ( B19 ), we finally obtain the first amendment N 1, in : where . APPENDIX C: SOLUTION FOR <inline-formula><tex-math id="TM0132" notation="LaTeX"></tex-math></inline-formula> BY AI To obtain a solution for the problem, we introduce the following dimensionless variables: , , , , and . Note that for such a problem, the diffusion coefficient κ can be rewritten in the following form: . This follows from the fact that and . Taking into account the system of ( 9 ) and restrict by a term N 0 in the first two equations of ( 11 ), we obtain a system of equations for finding the zero-order solution for each region of CR modulation: The last system can be rewritten in the following form: Then, the solution of the last system has the form To obtain the first amendments N 1, in , N 1, ihs , N 1, out , we insert ( C3 ) into ( 10 ): To derive j x , 1, in and j x ,
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