Travelling Waves Connected to Blood Flow and Motion of Arterial Walls

Dimitrova, Zlatinka I.; Vitanov, Nikolay K.

doi:10.1007/978-3-030-67227-0_12

Cited by 10 publications

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References 70 publications

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“…The list of examples of the application of the SEsM could continue. For more applications, see, for example, [ 460 , 463 , 467 , 468 , 469 , 470 , 484 , 485 , 487 , 488 , 508 , 509 , 510 ].…”

Section: Other Examples Of the Application Of The Sesmmentioning

confidence: 99%

“…About 13 years ago we started the development of a methodology for obtaining exact and approximate solutions of nonlinear differential equations. This methodology is now called the SEsM (Simple Equations Method) [ 460 , 461 , 462 , 463 , 464 , 465 , 466 , 467 , 468 , 469 , 470 , 471 ]. Elements of this methodology were used in our articles written decades ago [ 472 , 473 , 474 , 475 ].…”

Section: Short Introduction and Overview Of The Literaturementioning

confidence: 99%

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Simple Equations Method (SEsM): An Effective Algorithm for Obtaining Exact Solutions of Nonlinear Differential Equations

Vitanov

2022

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Exact solutions of nonlinear differential equations are of great importance to the theory and practice of complex systems. The main point of this review article is to discuss a specific methodology for obtaining such exact solutions. The methodology is called the SEsM, or the Simple Equations Method. The article begins with a short overview of the literature connected to the methodology for obtaining exact solutions of nonlinear differential equations. This overview includes research on nonlinear waves, research on the methodology of the Inverse Scattering Transform method, and the method of Hirota, as well as some of the nonlinear equations studied by these methods. The overview continues with articles devoted to the phenomena described by the exact solutions of the nonlinear differential equations and articles about mathematical results connected to the methodology for obtaining such exact solutions. Several articles devoted to the numerical study of nonlinear waves are mentioned. Then, the approach to the SEsM is described starting from the Hopf–Cole transformation, the research of Kudryashov on the Method of the Simplest Equation, the approach to the Modified Method of the Simplest Equation, and the development of this methodology towards the SEsM. The description of the algorithm of the SEsM begins with the transformations that convert the nonlinearity of the solved complicated equation into a treatable kind of nonlinearity. Next, we discuss the use of composite functions in the steps of the algorithms. Special attention is given to the role of the simple equation in the SEsM. The connection of the methodology with other methods for obtaining exact multisoliton solutions of nonlinear differential equations is discussed. These methods are the Inverse Scattering Transform method and the Hirota method. Numerous examples of the application of the SEsM for obtaining exact solutions of nonlinear differential equations are demonstrated. One of the examples is connected to the exact solution of an equation that occurs in the SIR model of epidemic spreading. The solution of this equation can be used for modeling epidemic waves, for example, COVID-19 epidemic waves. Other examples of the application of the SEsM methodology are connected to the use of the differential equation of Bernoulli and Riccati as simple equations for obtaining exact solutions of more complicated nonlinear differential equations. The SEsM leads to a definition of a specific special function through a simple equation containing polynomial nonlinearities. The special function contains specific cases of numerous well-known functions such as the trigonometric and hyperbolic functions and the elliptic functions of Jacobi, Weierstrass, etc. Among the examples are the solutions of the differential equations of Fisher, equation of Burgers–Huxley, generalized equation of Camassa–Holm, generalized equation of Swift–Hohenberg, generalized Rayleigh equation, etc. Finally, we discuss the connection between the SEsM and the other methods for obtaining exact solutions of nonintegrable nonlinear differential equations. We present a conjecture about the relationship of the SEsM with these methods.

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Section: Other Examples Of the Application Of The Sesmmentioning

confidence: 99%

Section: Short Introduction and Overview Of The Literaturementioning

confidence: 99%

Simple Equations Method (SEsM): An Effective Algorithm for Obtaining Exact Solutions of Nonlinear Differential Equations

Vitanov

2022

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“…The blood circulation is a complex system that affects the entire body. In order to better understand the physiological and pathological behavior of the cardiovascular system, it is important to study the dynamics of blood flow in the arteries and the mechanical factors of blood flow [2]. Therefore, studies concentrate on hemodynamics, which examines the properties of arterial blood flow movement.…”

Section: Introductionmentioning

confidence: 99%

Simulation studies on the hemodynamic models for blood flow

İzgi

2024

International Journal of Mathematics and Computer in Engineering

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One of the important fluid flows is blood flow seen in hemodynamics, which is a vital process and carries many ingredients from one place to another in the interior of the body. Blood is a special suspension; it is a non-Newtonian fluid as the blood flow cannot be compressed due to the imbalance in strain force and velocity. The blood flow is modelled by various equations which are based on fundamental equations such as Korteweg-De Vries (KdV) equation, nonlinear Schrödinger type equations. Our aim in this study is that the solitary solutions of the blood flow models are obtained in explicit form via Bernoulli method which is one of the ansatz-based methods. The results will be an important role in improving the models and they will support the theoretical basis and diagnose some blood diseases.

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“…We note that with respect to presentation of the solution of the solved NPDE as a series of solution of the simplest equation by means of a balance equation, the MMSE is identical to the MSE developed by Kudryashov. Applications of MMSE for obtaining exact solutions of different non-linear differential equations can be found in [ 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 , 65 , 66 , 67 , 68 ]. In [ 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 , 65 ], the authors used ordinary differential equations of first order, such as the equation of Riccati, the equation of Bernoulli and the equation of Abel of the first kind as simplest equations.…”

Section: Introductionmentioning

confidence: 99%

“…Applications of MMSE for obtaining exact solutions of different non-linear differential equations can be found in [ 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 , 65 , 66 , 67 , 68 ]. In [ 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 , 65 ], the authors used ordinary differential equations of first order, such as the equation of Riccati, the equation of Bernoulli and the equation of Abel of the first kind as simplest equations. Ordinary differential equations of the second order, such as elliptic equations and equations, based on the function

are used as the simplest equations in [ 66 , 67 , 68 ].…”

Section: Introductionmentioning

confidence: 99%

Exact Travelling-Wave Solutions of the Extended Fifth-Order Korteweg-de Vries Equation via Simple Equations Method (SEsM): The Case of Two Simple Equations

Nikolova

2022

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We apply the Simple Equations Method (SEsM) for obtaining exact travelling-wave solutions of the extended fifth-order Korteweg–deVries (KdV) equation. We present the solution of this equation as a composite function of two functions of two independent variables. The two composing functions are constructed as finite series of the solutions of two simple equations. For our convenience, we express these solutions by special functions V, which are solutions of appropriate ordinary differential equations, containing polynomial non-linearity. Various specific cases of the use of the special functions V are presented depending on the highest degrees of the polynomials of the used simple equations. We choose the simple equations used for this study to be ordinary differential equations of first order. Based on this choice, we obtain various travelling-wave solutions of the studied equation based on the solutions of appropriate ordinary differential equations, such as the Bernoulli equation, the Abel equation of first kind, the Riccati equation, the extended tanh-function equation and the linear equation.

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Travelling Waves Connected to Blood Flow and Motion of Arterial Walls

Cited by 10 publications

References 70 publications

Simple Equations Method (SEsM): An Effective Algorithm for Obtaining Exact Solutions of Nonlinear Differential Equations

Simple Equations Method (SEsM): An Effective Algorithm for Obtaining Exact Solutions of Nonlinear Differential Equations

Simulation studies on the hemodynamic models for blood flow

Exact Travelling-Wave Solutions of the Extended Fifth-Order Korteweg-de Vries Equation via Simple Equations Method (SEsM): The Case of Two Simple Equations

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