On Exact Series Solution for Strongly Coupled Mixed Parabolic Boundary Value Problems

Algorithm 1

Solution of the homogeneous problem with homogeneous conditions (10)–(13).

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

Result: 𝑢(𝑥,𝑡).

(1) Verify that matrix 𝐴 satisfies (2).

(2) Verify 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 in (5).

(5) Determine matrices 𝐴₂ and 𝐵₂ defined in (7).

(6) Consider the following cases:

 (i) Case1: Condition (27) is fulfilled, that is, matrices 𝐵₁ and 𝐴₂ have a common eigenvector 𝑤≠0 associated with

  eigenvalues 0∈𝜎𝐵₁ and 𝑎₂∈𝜎𝐴₂. In this case we will continue with step (7).

 (ii) Case2: Condition (27) is not fulfilled. In this case the algorithm is not completed because it is not possible to find the

  solution of (10)–(13) for the given data.

(7) Take 0∈𝜎𝐵₁ and determine 𝑎₂∈𝜎𝐴₂⋂ℝ, and vector 𝑤≠0 when 𝑤∈𝐾𝑒𝑟𝐵₁⋂Ker𝐴₂ in a way that:

 (i) Conditions (59) fulfilled, that is:

 (1.1) Ker𝐵₁⋂Ker𝐴₂−𝑎₂𝐼 is an invariant subspace respect to matrix 𝐴.

 (1.2) 𝑓(𝑥)∈Ker𝐵₁⋂Ker𝐴₂−𝑎₂𝐼, ∀𝑥∈[0,1].

 (ii) The vectorial function 𝑓(𝑥) fulfils (50), that is:

 (1.3) 𝑓∈𝒞²([0,1]).

 (1.4) 𝑓(0)=0.

 (1.5) 𝑎₂𝑓(1)+𝑓^′(1)=0.

If these conditions are not satisfied, go back over step (6) of Algorithm (1) and discard the value taken for 𝐚_𝟐 .

(8) Determine the positive solutions of (30) and determine ℱ defined in (34).

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

(10) Build block matrix 𝐺_𝜆𝑘(𝜌₀) defined in (39).

(11) Determine 𝜆_𝑘∈ℱ so that rank𝐺_𝜆𝑘(𝜌₀)<𝑚.

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

(13) Determine 𝛼 given in (52).

(14) Determine vectors 𝐶(𝜆_𝑛) defined in (55).

(15) Determine the series solution 𝑢(𝑥,𝑡) of problem (10)–(13) defined in (57).