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The paper concerns a nonlinear second-order system of coupled PDEs, having the principal part in <i>divergence</i> form and subject to in-homogeneous dynamic boundary conditions, for both <inline-formula><math display="inline"><semantics><mrow><mi>θ</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>φ</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Two main topics are addressed here, as follows. First, under a certain hypothesis on the input data, <inline-formula><math display="inline"><semantics><msub><mi>f</mi><mn>1</mn></msub></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><msub><mi>f</mi><mn>2</mn></msub></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><msub><mi>w</mi><mn>1</mn></msub></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><msub><mi>w</mi><mn>2</mn></msub></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><msub><mi>θ</mi><mn>0</mn></msub></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><msub><mi>α</mi><mn>0</mn></msub></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><msub><mi>φ</mi><mn>0</mn></msub></semantics></math></inline-formula>, and <inline-formula><math display="inline"><semantics><msub><mi>ξ</mi><mn>0</mn></msub></semantics></math></inline-formula>, we prove the well-posedness of a solution <inline-formula><math display="inline"><semantics><mrow><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>φ</mi><mo>,</mo><mi>ξ</mi></mrow></semantics></math></inline-formula>, which is <inline-formula><math display="inline"><semantics><mrow><mfenced separators="" open="(" close=")"><mi>θ</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>α</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mfenced><mo>∈</mo><msubsup><mi>W</mi><mi>p</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>W</mi><mi>p</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mo>Σ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mfenced separators="" open="(" close=")"><mi>φ</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>ξ</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mfenced><mo>∈</mo><msubsup><mi>W</mi><mi>ν</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>W</mi><mi>p</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mo>Σ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>ν</mi><mo>=</mo><mo movablelimits="true" form="prefix">min</mo><mo>{</mo><mi>q</mi><mo>,</mo><mi>μ</mi><mo>}</mo></mrow></semantics></math></inline-formula>. According to the new formulation of the problem, we extend the previous results, allowing the new mathematical model to be even more complete to describe the diversity of physical phenomena to which it can be applied: interface problems, image analysis, epidemics, etc. The main goal of the present paper is to develop an iterative scheme of fractional-step type in order to approximate the unique solution to the nonlinear second-order system. The convergence result is established for the new numerical method, and on the basis of this approach, a conceptual algorithm, <b>alg-frac_sec-ord_</b><tt>u+varphi</tt><b>_dbc,</b> is elaborated. The benefit brought by such a method consists of simplifying the computations so that the time required to approximate the solutions decreases significantly. Some conclusions are given as well as new research topics for the future.