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tokens: the zero-order solution: The last system of equations can be rewritten in the following form: Then, the solution of the last system has the form To obtain the first and next amendments, we use a form such as N i = C η −α Φ i ( x ), and when we insert N 0 into ( 11 ), we will obtain an equation for N 1 : If we expand the exponent in the last equation in a Taylor series, we obtain Here, the first condition ( 12 ) was applied, i.e. the integration was carried out from 0 to x . We will seek a solution of the last equation in the following form: After an expansion of the exponent in a series followed by integration, we obtain The last condition ( 12 ) for finding B was used, i.e. Φ 1 ( x 0 ) = 0, and then B has the following form: And, as a result, the first amendment is When we perform similar actions in order to find the second amendment, we will obtain An exact solution for this problem was initially obtained by Dolginov Toptygin ( 1967 ). In our notation, it has the following form: where F ( a, b ; x ) is the confluent hypergeometrical function (Bateman Erdelyi 1953 ). In particular, from the property of the confluent hypergeometrical function, it follows that for the simpler case when α = 3, the exact solution ( A11 ) has the form APPENDIX B: κ = <italic>k</italic><sub>0</sub>η To obtain a solution for the problem, we introduce the following dimensionless variables: and . Then, taking into account ( 6 ) and ( 9 ), we obtain a system of questions for finding the zero-order solution: The last system can be rewritten in the following form: Then, the solution of the last system has the form After inserting N 0 into ( 11 ), we will obtain an equation for N 1 : After differentiating the left-hand side of the last equation, we obtain where Here, the limits of integration were selected as being from 0 to x in consequence of applying the first condition ( 12 ). Now, we consider the integral I separately. If we define , then the expression for I can be rewritten in the following form: Whereas N 0 has the form ( B3 ), then, taking into account the second condition ( 12 ), it is necessary to search for an N 1 of the following form: And, as a result, the first amendment takes the following form: It should be noted that N 1 is not defined for α = 0, 12. APPENDIX C: κ = <italic>k</italic><sub>0</sub>βη 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 ( 6 ) and ( 9 ), we obtain a system of equations for finding the zero-order solution: The last system can be rewritten in the form Then, the solution of the last system has the form After inserting N 0 into ( 11 ), we will obtain an equation for N 1 : After introducing the variable: and simplifying the last equation, we obtain where . Note that the limits of integration in K, L, M were selected as being from 0