On Exact Series Solution of Strongly Coupled Mixed Parabolic Problems

Algorithm 1

Solution of the homogeneous problem with homogeneous conditions (1)–(4).

Input data: 𝐴,𝐴₁,𝐴₂,𝐵₁,𝐵₂∈ℂ^𝑚×𝑚, 𝑓(𝑥)∈ℂ^𝑚.

Result: 𝑢(𝑥,𝑡).

(1) Check that matrix 𝐴 satisfies (5).

(2) Check that matrices 𝐴_𝑖,𝐵_𝑖∈ℂ^𝑚×𝑚,𝑖∈{1,2} are singular, and check that the block matrix

    𝐴₁𝐵₁𝐴₂𝐵₂ is regular.

(3) Determine a number 𝜌₀∈ℝ so that the matrix pencil 𝐴₁+𝜌₀𝐵₁ is regular.

(4) Determine matrices 𝐴₁ and 𝐵₁ defined by (10).

(5) Determine matrices 𝐴₂ and 𝐵₂ defined by (11).

(6) Consider the following cases:

  (i) Case  1. Condition (13) holds, that is, matrices 𝐵₁ and 𝐵₂ have a common eigenvector 𝑣≠0 associated

   with eigenvalues 𝑏₁∈𝜎𝐵₁−{0} and 𝑏₂∈𝜎𝐵₂. In this case continue with step (7).

  (ii) Case  2. Condition (13) does not hold. In this case the algorithm stops because it is not possible to

   find the solution of (1)–(4) for the given data.

(7) Determine 𝑏₁∈𝜎𝐵₁, 𝑏₁≠0, 𝑏₂∈𝜎𝐵₂ and vector 𝑣≠0 verifying

   𝑣∈Ker𝐵₁−𝑏₁𝐼∩Ker𝐵₂−𝑏₂𝐼 such that:

  (i) Conditions (53) hold, that is:

  1.1: Ker𝐵₁−𝑏₁𝐼∩Ker𝐵₂−𝑏₂𝐼 is an invariant subspace respect matrix 𝐴.

  1.2: 𝑓(𝑥)∈Ker𝐵₁−𝑏₁𝐼∩Ker𝐵₂−𝑏₂𝐼, ∀𝑥∈[0,1].

    (ii) Conditions (14) hold, that is:

  1.3: 1−𝑏₂+𝜌₀𝑏₁𝑏₂1−𝜌₀𝑏₁𝑏₁∈ℝ, 𝑏₁𝑏₂∈ℝ.

    (iii) The vectorial function 𝑓(𝑥) satisfies (42), that is:

  1.4: 𝑓∈𝒞²([0,1]).

  1.5: 1−𝜌₀𝑏₁𝑓(0)+𝑏₁𝑓^′(0)=0.

  1.6: −1−𝑏₂+𝜌₀𝑏₁𝑏₂𝑏₁𝑓(1)+𝑏₂𝑓^′(1)=0.

If these conditions are not satisfied, return to step (6) of Algorithm 1 discarding the values

  taken for 𝑏₁ and 𝑏₂.

(8) Determine the positive solutions of (16) and determine ℱ defined by (27).

(9) Determine degree 𝑝 of minimal polynomial of matrix 𝐴.

(10) Building block matrix 𝐺𝜌₀,𝑏₁,𝑏₂,𝜆_𝑘 defined by (31).

(11) Determine 𝜆∈ℱ so that rank 𝐺𝜌₀,𝑏₁,𝑏₂,𝜆_𝑘<𝑚.

(12) Include the eigenvalue 𝜆=0 if 1∈𝜎−𝐴₂𝐴₁.

(13) Determine 𝛼 given by (44).

(14) Determine vectors 𝐶(𝜆_𝑛) defined by (47).

(15) Determine functions 𝑢(𝑥,𝑡,𝜆_𝑛) defined by (34).

(16) Determine the series solution 𝑢(𝑥,𝑡) of problem (1)–(4) defined by (49).