Simple Equations Method (SEsM): An Effective Algorithm for Obtaining Exact Solutions of Nonlinear Differential
1Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria.
Entropy (Basel, Switzerland)
|November 24, 2022
Summary
The Simple Equations Method (SEsM) provides exact solutions for complex nonlinear differential equations, crucial for modeling phenomena like epidemic waves. This review details the SEsM
Area of Science:
- Mathematical Physics
- Nonlinear Dynamics
- Computational Mathematics
Background:
- Exact solutions of nonlinear differential equations are vital for understanding complex systems.
- Existing methods like Inverse Scattering Transform and Hirota's method have limitations.
- The review contextualizes the Simple Equations Method (SEsM) within the broader landscape of solution methodologies.
Purpose of the Study:
- To provide a comprehensive overview of the Simple Equations Method (SEsM) for obtaining exact solutions.
- To detail the algorithmic steps and underlying principles of the SEsM.
- To demonstrate the versatility and applicability of the SEsM across various scientific domains.
Main Methods:
- The Simple Equations Method (SEsM) transforms complex nonlinearities into simpler, solvable forms.
- Utilizes composite functions and a 'simple equation' (e.g., Bernoulli, Riccati) as a core component.
- Connects SEsM to established techniques like Hopf-Cole transformation and Kudryashov's Method of the Simplest Equation.
Main Results:
- SEsM successfully generates exact solutions for numerous nonlinear differential equations.
- Demonstrates applications in modeling epidemic waves (e.g., COVID-19) via the SIR model.
- Introduces a special function encompassing trigonometric, hyperbolic, and elliptic functions.
Conclusions:
- The SEsM is a powerful and adaptable methodology for finding exact solutions to nonlinear differential equations.
- It offers a unified approach, defining special functions and connecting to other solution techniques.
- Further research is suggested regarding the relationship between SEsM and methods for nonintegrable equations.
Keywords:
COVID-19Faa di Bruno formula for derivatives of a composite functionG’/G methodJacobi elliptic function expansion methodKorteweg-de Vries equationOlver equationSIR model of epidemic spreadingauxiliary equation methoddifferential equation of Bernoullidifferential equation of Riccatiequation of Burgers–Huxleyequation of fisherexact solutionsexp-function methodf-expansion methodfirst integral methodgeneral projective Riccati equations methodgeneralized Rayleigh equationgeneralized equation of Camassa–Holmgeneralized equation of Swift–Hohenberghomogeneous balance methodmethod of Hirotamethod of the inverse scattering transformmodified method of the simplest equationmodified simple equation methodnonlinear Schrödinger equationnonlinear differential equationssimple equations methodsolitonstanh methodtrial function method
Related Concept Videos
01:21Linear Approximation in Time Domain
114
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
114
01:26Linear Approximation in Frequency Domain
123
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
123
01:15Kinematic Equations: Problem Solving
12.6K
When analyzing one-dimensional motion with constant acceleration, the problem-solving strategy involves identifying the known quantities and choosing the appropriate kinematic equations to solve for the unknowns. Either one or two kinematic equations are needed to solve for the unknowns, depending on the known and unknown quantities. Generally, the number of equations required is the same as the number of unknown quantities in the given example. Two-body pursuit problems always require two...
12.6K
02:20The Small x Assumption
46.4K
If a reaction has a small equilibrium constant, the equilibrium position favors the reactants. In such reactions, a negligible change in concentration may occur if the initial concentrations of reactants are high and the Kc value is small. In such circumstances, the equilibrium concentration is approximately equal to its initial concentration. This estimation can be used to simplify the equilibrium calculations by assuming that some equilibrium concentrations are equal to the initial...
46.4K
01:24Difference Equation Solution using z-Transform
351
The z-transform is a powerful tool for analyzing practical discrete-time systems, often represented by linear difference equations. Solving a higher-order difference equation requires knowledge of the input signal and the initial conditions up to one term less than the order of the equation.
The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the...
The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the...
351
01:21Chemical Equilibria: Systematic Approach to Equilibrium Calculations
788
Equilibrium calculations for systems involving multiple equilibria are often complex. For example, to calculate the solubility of a sparingly soluble salt in an aqueous solution in the presence of a common ion, one must consider all the equilibria in this solution. Calculations for these systems can be complicated and tedious, so a systematic approach with a series of steps is often helpful. The process is detailed below.
The first step is to identify all the chemical reactions involved, The...
The first step is to identify all the chemical reactions involved, The...
788