Nonlinear Dynamics of the KdV-B Equation and Its Biomedical Applications

In recent years there is an incremental degree of bridging open questions in biomechanics with the help of applied mathematics and nonlinear analysis. Recent advancements concerning the cardiac dynamics pose important questions about the cardiac waveform. A governing equation, namely the KdV-B equation (Korteweg–de Vries–Burgers), 1 ∂ u ∂ t + γ u ∂ u ∂ x − α ∂ 2 u ∂ x 2 + β ∂ 3 u ∂ x 3 = 0 , u = u ( t , x ) , α , β , γ ∈ ℝ , $$\displaystyle {} \dfrac {\partial u}{\partial t}+\gamma u\dfrac {\partial u}{\partial x}-\alpha \dfrac {\partial ^{2}u}{\partial x^{2}}+\beta \dfrac {\partial ^{3}u}{\partial x^{3}}=0, \ \ \ u=u(t,x), \ \ \ \alpha ,\,\beta ,\,\gamma \in \mathbb {R} , $$ is a partial differential equation utilized to answer several of those questions. The cardiac dynamics mathematical model features both solitary and shock wave characteristics due to the dispersion and dissipation terms, as occurring in the arterial tree. In this chapter a focus is given on describing cardiac dynamics. It is customarily difficult to solve nonlinear problems, especially by analytical techniques. Therefore, seeking suitable solving methods, exact, approximate or numerical, is an active task in branches of applied mathematics. The phase plane of the KdV–B equation is analyzed and its qualitative behavior is derived. An asymptotic expansion is presented and traveling wave solutions under both shock and solitary profiles are sought. Numerical solutions are obtained for the equation, by means of the Spectral Fourier analysis and are evolved in time by the Runge–Kutta method. This whole analysis provides vital information about the KdV–B equation and its connection to cardiac hemodynamics. The applications of KdV–B, presented in this chapter, highlight its essence to human hemodynamics.