The simplest equations with variable coefficients are considered in this research. The purpose of this study is to extend the procedure for solving the nonlinear differential equation with variable coefficients. In this case, the generalized Riccati equation is solved and becomes a basis to tackle the nonlinear differential equations with variable coefficients. The method shows that Jacobi and Weierstrass equations can be rearranged to become Riccati equation. It is also important to highlight that the solving procedure also involves the reduction of higher order polynomials with examples of Korteweg de Vries and elliptic-like equations. The generalization of the method is also explained for the case of first order polynomial differential equation.
- the simplest equation
- Riccati equation
- nonlinear differential equations
- reduction of polynomial
Despite the advent of supercomputers in numerical methods, increasing activities are devoted to solving nonlinear differential equations by analytical method in recent years [1, 2, 3]. Analytical solutions have their own importance concerning the physical phenomena as they are often pave the way to the construction of right theory . Many methods have been proposed concerning this important problem and generally, for the problems with constant coefficients. One of the useful methods is the method of simplest Equation  or for some authors, the auxiliary Equations . The method is built by the utilization of the first integral of simplest nonlinear differential equations, such as Bernoulli and Riccati Equations . The method had produced many new solutions of the considered nonlinear differential equations, generally with constant coefficients [8, 9].
For the more general cases, we have found that the method can be extended such as involving the solution of the nonlinear differential equations with variable coefficients. The nature of variable coefficients often arises in the equation describing the heterogenous media and composites  or in other cases are produced by the coordinate transformation of the partial differential Equations . Those two categories are developed rapidly in recent years with the capacity of high-speed super computers which sufficient for computing nonlinear problem with complex geometries [12, 13], as sometimes desired by engineering design activities. The role of analytical solutions is as a benchmark to validate the computer algorithm with simpler geometries as it is usually performed .
In this chapter, the solution method of the simplest equations is different from the cases of constant coefficients except, on Bernoulli equation. Hence, we will start from Riccati equation instead of Bernoulli equation as the simplest equation to highlight the novelty of the procedure. The method is then followed by examples and conclusion.
2. The first integral of the simplest equations with variable coefficients
2.1 Riccati equation
Consider the Riccati equation with variable coefficients as follows,
Let and the above equation can be rearranged as,
and is separated as
The solutions for and are
The relation for is thus,
Without loss of generality, suppose that and thus the above relation is performed as,
Rearrange Eq. (5) and integrate once,
The solution for is then,
where the coefficients will be determined later from the substitution into the considered nonlinear differential equations.
2.2 The Jacobi and Weierstrass equations
It is interesting to know that other simplest equations can also be rearranged into the Riccati-type equations. The famous examples are Jacobi  and Weierstrass Equations , which can solve a large class of nonlinear differential equations. Let us consider Jacobi type equation with variable coefficients,
and the Weierstrass equation as follows,
Here, the reader should not be confused by the coefficients which represent different functions with the same index. Take and the Weierstrass equation becomes Jacobi equation which admits the similar method of solution.
Concerning the search for obtaining solution of (7) and (8), the balancing principle suggests the substitution of the first order series as in the following,
Performing the Riccati equation into (9) and we generate the following expression,
The coefficients of polynomial are then related with the coefficients in (9) in order to determine as functions of the known and as follows,
Hence, the first equation gives,
and the second equation is then,
The next relation produces,
which the roots will determine the solution for as functions of , or in simple unknown variable. The step now is to find the polynomial expression for from the last relation of (10) as,
Therefore, the last equation gives the expression for as polynomial equation of higher order, and the generated polynomial is,
In this case the higher order polynomial will be solved by reducing the order.
3. Reduction of higher order polynomial
Consider the sixth order polynomial equation as in the following,
First, multiply the above equation with the function and rearranged as,
where . The polynomial equation is cut as in the following,
Note that, the coefficients in this section is different from the previous section. Expanding for the new coefficients,
Hence, the relation for coefficients is,
The fifth coefficient relation is rearranged as,
and the roots are,
Also, the last relation is rewritten as,
Note that performing (19) into (21) will remove and . Thus, it is necessary to take other relation, i.e. , which will produce the cubic equation as follows,
which has the roots as,
Substituting back into to get,
Therefore, is also determined by (24) and so all the coefficients, . The polynomial equation of sixth order is then re-expressed as,
as reduced into the quartic equation the roots can be obtained by radical solution.
The procedure described by (17–25) can be applied and iterated into (16) until the polynomial equation of is reduced into quartic equation. Hence, all the coefficients for Riccati equation the first order series, i.e. are determined and produce the solution as,
Thus, following the method explained by (2–6) and (17–25), we have arrived at the solution of Jacobi and Weierstrass equations with variable coefficients.
4. Solution examples
4.1 The elliptic-like equation
As an application, consider the elliptic-like equation with forcing function,
The balance principle suggests that the solution should be in the form,
Substituting into (27) will reproduce the following expression,
The next step is to differentiate the Riccati equation once,
Substituting into Eq. (29) and it will produce the polynomial equation as in the following,
or the next step is to relate the coefficients as,
In this case, the first equation gives,
For the second equation,
Thus, provide the expression for as,
The third and fourth equations produce,
Substituting (32) into (31),
Replace with (10),
which then solves regardless of . In this case, we take as the chosen fundamental variable and the resulted coefficients, depend on and with the known coefficients . Therefore, the solution of (27) is generated as,
4.2 Korteweg de Vries equation
The next example is for the Korteweg de Vries type equation,
The balancing principle with application of Riccati equation will determined the ansatz,
Performing into (35) will produce,
Performing the Riccati equation into (37) will produce the following polynomial,
From this step on, there is a little hope to solve all the coefficients as they are equal to zero. As it has also to be reduced, it is important to note that the problem of reduction here is different from the case of Jacobi equation since all the coefficients are in principle solvable in algebraic form. In this case, it is not practical to reduce the fifth order polynomial as the even highest power, i.e., as a tenth order polynomial equation. The calculation will become too tedious as the detail expression is needed in the reduced polynomial equation. The next sub section will illustrate the reduction of an odd highest power polynomial equation.
4.3 Reduction of fifth order polynomial
Consider Eq. (38) as follows,
Multiply by the function, and rearrange,
where, . Rearranged Eq. (39) as given by,
Expanding the all the coefficients as,
Relate the coefficients as in the following,
The fifth equation of (41) gives the roots as,
Moving to the last equation, the functions and disappear from the operation. In this case we will consider the test function, , and will perform as,
Substituting for , the expression for is,
Substitute back to, as follows
which then solves , , and thus generates all the coefficients of . The polynomial is then rewritten as,
which is reduced as,
Eq. (46) dictates that the relations, will satisfy for the solution. Hence, the coefficients are then,
The first equation gives,
The second equation is rewritten as,
The solution for is then,
The third equation will produce,
Take the expression for as,
The fourth relation of (47) will generate,
which then produce the solution of .
Note that and are the chosen fundamental variables and according to (48–51) and with the known coefficients , they will define, . Therefore, the solution of Korteweg de Vries equation is generated as,
Since only a few of the considered equation has a special polynomial to be solved by equating all the variable coefficients to zero, it is important to note that the reduction of polynomial order would be an important step. Solving all coefficients to zero often be an obstacle because the difficulty would be the same or even more than the original nonlinear ODEs. In this case, the reduction of polynomial manipulates and reduces the need for solving all coefficients.
with . The solution of (53) is then,
The definition for is determined by substituting the polynomial solution, into (54) as in the following,
Rearranging the above equation as,
Eq. (55) is a first order ODE in and can be easily solved, which then prove that without establishing the explicit expression for variable coefficients.
5. Generalized method
In this section, the method of solution to the Riccati equation is extended for the class of the first order polynomial differential equation as,
The above equation can be always re-expressed as,
which the coefficients will be reformulated as,
In this case, we will always obtain the new coefficients . Proceeding into the other equations andmultiply the equation by the function , to get,
where . Then, all the new coefficients in will be determined. The step is now to solve the Riccati equation. Let , the equation can be rearranged as,
and is separated as,
The solutions for and are,
The relation for is thus,
Without loss of generality, suppose that and the above relation is performed as,
Rearrange the above equation as,
Let and integrate the above equation to get,
The solution for is then reduced into the solution of the polynomial equation. Thus, let , where is the expression from the solution of the resulting polynomial equation which is similar to (38). The expression for can be determined by the inverse method as in (53–55) for the first order polynomial differential Eq. (56).
In this chapter, we propose the method of the simplest or the auxiliary equation to solve the nonlinear differential equation with variable coefficients. The method is based on the solution of the generalized Riccati equation as the simplest equation. It is found that the other known simplest equations, i.e., Jacobi and Weierstrass equation, are also solved by the Riccati equation. The applications with the variable coefficients elliptic-like and Korteweg de Vries equations show that the problem of solving nonlinear differential equations with variable coefficients are simplified, especially by the reduction of the resulting polynomial equation in solving the Korteweg de Vries equation. The generalization of the method is also derived in detail.
Conflict of interest
Authors declare that there is no conflict of interest.
Khan K, Akbar MA, Koppelaar H. Study of Coupled Nonlinear Partial Differential Equations for Finding Exact Analytical Solutions. R. Soc. open Sci. 2015; 2; 140406. DOI:10.1098/rsos.140406
Dubey RS, Goswami P. Analytical Solution of the Nonlinear Diffusion Equation. The European Physical Journal Plus. 2018; 133; 183. DOI: 10.1140/epjp/i2018-12010-6
Polyanin AD. Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions Mathematics. 2019; 7; 386. DOI: 10.3390/math7050386
Saengow C, Giacomin AJ, Kolitawong C. Exact Analytical Solution for Large-Amplitude Oscillatory Shear Flow. Macromolecular: Theory and Simulations. 2015; 24; 4. DOI: 10.1002/mats.201400104
Vitanov NK. Modified Method of Simplest Equation for Obtaining Exact Solutions of Nonlinear Partial Differential Equations: History, Recent Developments of the Methodology and Studied Class of Equations. Journal of Theoretical and Applied Mechanics, Sofia. 2019;49; 107–122. DOI: 10.7546/JTAM.49.19.02.02
Liu Q, Zhang R, Yang L, Song J. A New Model for Nonlinear Rossby Waves and Some of its Solutions. Phys. Lett. A. 2019; 383; 514–525. DOI:10.1016/j.physleta.2018.10.052
Vitanov NK. Modified Method of Simplest Equation: Powerful Tool for Obtaining Exact and Approximate Traveling-Wave Solutions of Nonlinear PDEs. Commun. Nonlinear Sci. Numer. Simulat. 2011; 16; 1176–1185. DOI:10.1016/j.cnsns.2010.06.011
Zhao YM, He YH, Long Y. The Simplest Equation Method and Its Application for Solving the Nonlinear NLSE, KGZ, GDS, DS, and GZ Equations. Journal of Applied Mathematics. 2013; 960798. DOI: 10.1155/2013/960798
Kudryashov NA, Loguinova NB. Extended Simplest Equation Method for Nonlinear Differential Equations. Applied Mathematics and Computation. 2008; 205; 396–402. DOI: 10.1016/j.amc.2008.08.019
Decarlis A, Jaeger M, Martin R. Determination of the Effective Thermal Conductivity Tensor of Heterogeneous Media Using a Self-Consistent Finite Element Method: Application to the Pseudo-Percolation Thresholds of Mixture Containing Nonspherical Inclusions. Journal of Heat Transfer. 2000; 122; 171–175. DOI: 10.1115/1.521451
Qi C, Huang S. Variable-Coefficient Exact Solutions for Nonlinear Differential Equations by a New Bernoulli Equation-Based Subequation Method. Mathematical Problems in Engineering. 2013; 3; 923408. DOI: 10.1155/2013/923408
Tao WQ, Cheng YP, Lee TS. 3D Numerical Simulation on Fluid Flow and Heat Transfer Characteristics in Multistage Heat Exchanger with Slit Fins. Heat and Mass Transfer. 2007; 44; 125–136. DOI:10.1007/s00231-006-0227-2
Tabani M, Babayan N, Pouyaei A. Optimization of PV/Wind/Battery Stand-Alone System, Using Hybrid FPA/SA Algorithm and CFD Simulation, Case Study: Tehran. Energy Conversion and Management. 2015; 106; 644–659. DOI:10.1016/j.enconman.2015.10.011
Barbato D, Berselli LC, Grisanti CR. Analytical and Numerical Results for the Rational Large Eddy Simulation Model. J. Math. Fluid Mech. 2007; 9; 44–74. DOI:10.1007/s00021-00-0191-0
Yu J, Wang DS, Sung Y, Wu S. Modified Method of Simplest Equation for Obtaining Exact Solutions of the Zakharov-Kuznetsov Equation, the Modified Zakharov-Kuznetsov Equation, and their Generalized Forms. Nonlinear Dynamics. 2016; 85; 2449–2465. DOI:1007/s11071–016–2837-7
Kudryashov NA. Exact Solutions of the Equation for Surface Waves in a Convecting Fluid. Applied Mathematics and Computation. 2019; 344–345; 97–106. DOI:10.1016/j.amc.2018.10.005