Exact Solutions of Non-Linear Evolution Models in Physics and Biosciences Using the Hyperbolic Tangent Method
Abstract
:1. Introduction
2. Preliminaries
2.1. Travelling Wave Solutions
2.2. The Hyperbolic Tangent Method (Tanh Method)
- Suppose one needs to determine solitary wave solutions to a non-linear partial differential equation of the formThe solution to Equation (1) is proposed to be a polynomialA travelling wave solution requires the coordinates: and , where represents the, localized, wave solution which travels with a velocity c and wave number k. Without loss of generality, we define . Consequently, the PDEs are transformed into ODEs. That is, for example the first and second order, partial derivatives with respect to time and space become
- The central step is the introduction of as a new independent variable and the corresponding derivatives are then changed as follows:Next, the degree of the polynomial (2) is determined by equating every two possible highest exponents in the equation to get a linear system for N and then that system is solved rejecting any solution N which is not a positive integer.
- Determining the degree of the polynomial and coefficients . Solving the non-linear system is the most involving step. So the following assumptions are made:
- (i)
- All parameters in the problem are considered strictly positive. If some parameters are zero, we must calculate N again because the polynomial might have changed.
- (ii)
- The coefficient of the highest power of Y term must be non-zero
- (iii)
- The wave number k is assumed to be positive.
- Substitute the solutions for the coefficients and parameters into the original equation.
- Korteweg-de Vries (KdV) equationThe KdV equation is one of the most famous non-linear partial differential equations. It was derived in fluid mechanics to describe shallow water waves in a rectangular channel. The equation is of the form:By setting , the partial derivatives of Equation (8) are transformed intoNext, we introduce and as previously derived, one obtainsFor the first term in Equation (11), we haveThe highest power of Y in the second term is . For the third term in (11), we start from the inner, right, derivative to the outer, left, derivative such thatThereforeWe can notice that the highest power of Y in the third term is . Taking the highest possible exponent of Y, we have implying that . Thus . The solution, , is therefore of the form . Substituting this solution into Equation (11), one obtains where and . The values of k and c are arbitrary. We can convince ourselves that
- Burgers equationThe Burgers equation is another famous non-linear PDE used in modelling various phenomena in applied mathematics. The positive parameter a is a diffusion constant which represents the dissipative effect of U. In order to determine closed form solutions of Burgers equation, the same schematic outline in the previous example is followed. Firstly, Equation (12) is transformed intoIntroducing and yieldsAfter substituting the derivatives of the polynomial into (14) and balancing the highest powers of Y, we arrive at in the second term and third term . Hence and therefore in this example . Hence the polynomial solution takes the formSubstituting Equation (15) into (14), the solution obtained is where The constants remain arbitrary and the values are greater than zero. Lastly, we need to find the value of and require that the solution vanishes for . So we let such that since . ThusThis is a well known shock wave solution for burgers equation [22].
3. Travelling Wave Solutions to Selected Models
3.1. FitzHungh-Nagumo Equation
3.2. Korteweg-de Vries-Burgers Equation
3.3. Melanoma Model
3.4. Microbial Growth Model
3.5. Tumour-Immune Interaction Model
4. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
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Malinzi, J.; Quaye, P.A. Exact Solutions of Non-Linear Evolution Models in Physics and Biosciences Using the Hyperbolic Tangent Method. Math. Comput. Appl. 2018, 23, 35. https://doi.org/10.3390/mca23030035
Malinzi J, Quaye PA. Exact Solutions of Non-Linear Evolution Models in Physics and Biosciences Using the Hyperbolic Tangent Method. Mathematical and Computational Applications. 2018; 23(3):35. https://doi.org/10.3390/mca23030035
Chicago/Turabian StyleMalinzi, Joseph, and Paul Amarh Quaye. 2018. "Exact Solutions of Non-Linear Evolution Models in Physics and Biosciences Using the Hyperbolic Tangent Method" Mathematical and Computational Applications 23, no. 3: 35. https://doi.org/10.3390/mca23030035