On the other hand, by Green's function method, a unique solution of the problem 1. Thus, 1. It can easily be verified that the homogeneous problem associated with 1. Therefore, by Green's function method, the solution of 1. Theorem 2. If a,3 e C2[0,1] are assumed to be lower and upper solutions of 1. Assume that a,fi e C2[0,1] are, respectively, lower and upper solutions of 1. For the sake of contradiction, let u have a positive maximum at some t0 e [0,1].
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On the other hand, in view of the decreasing property of f t, x, y in x, we have. Ai the functions a, 5 e C2[0,1] are, respectively, lower and upper solutions of 1. Furthermore, there exists a constant N depending on a, fi, and Nagumo function h such that. Thus, any solution x of 3. Therefore, by Taylor's theorem, we obtain. Thus, a,fi are lower and upper solutions of 3.
Hence by Theorems 2. Note that the uniqueness of the solution follows by Theorem 2.
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Using 3. In a similar manner, it can be shown by using A1 , 3. Also, in view of 3. Integrating 3. Also, the assumptions of Theorem 3. Thus, the conclusion of Theorem 3. The author is grateful to the referees and professor G.
Infante for their valuable suggestions and comments that led to the improvement of the original paper. Ladde, V. Lakshmikantham, and A. Nieto, Y. Jiang, and Y. Vatsala and J. Yang, "Monotone iterative technique for semilinear elliptic systems," Boundary Value Problems, vol. Drici, F. McRae, and J. Jiang, J. Nieto, and W.
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Zuo, "On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations," Journal of Mathematical Analysis and Applications, vol. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method.
There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.
Nonlinearity exists everywhere and, in general, nature is nonlinear. Nonlinear evolution partial differential equations arise in many fields of science, particularly in physics, engineering, chemistry, finance, and biological systems.
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They are widely used to describe complex phenomena in various fields of sciences, such as wave propagation phenomena, fluid mechanics, plasma physics, quantum mechanics, nonlinear optics, solid state physics, chemical kinematics, physical chemistry, population dynamics, financial industry, and numerous areas of mathematical modeling. The development of both numerical and analytical methods for solving complicated, highly nonlinear evolution partial differential equations continues to be an area of interest to scientists whose research aim is to enrich deep understanding of such alluring nonlinear problems.
Innumerable number of methods for obtaining analytical and approximate solutions to nonlinear evolution equations have been proposed. Some of the analytical methods that have been used to solve evolution nonlinear partial differential equations include Adomian's decomposition method [ 1 — 3 ], homotopy analysis method [ 4 — 7 ], tanh-function method [ 8 — 10 ], Haar wavelet method [ 11 — 13 ], and Exp-function method [ 14 — 16 ]. Several numerical methods have been used to solve nonlinear evolution partial differential equations.
These include the explicit-implicit method [ 17 ], Chebyshev finite difference methods [ 18 ], finite difference methods [ 19 ], finite element methods [ 20 ], and pseudospectral methods [ 21 , 22 ]. They may also be cumbersome to use as some involve manual integration of approximate series solutions and, hence, it is difficult to find closed solutions sometimes. On the other hand, some numerical methods may not work in some cases, for example, when the required solution has to be found near a singularity.
Certain numerical methods, for example, finite differences require many grid points to achieve good accuracy and, hence, require a lot of computer memory and computational time. Conventional first-order finite difference methods may result in monotonic and stable solutions, but they are strongly dissipative causing the solution of the strongly convective partial differential equations to become smeared out and often grossly inaccurate.
On the other hand, higher order difference methods are less dissipative but are prone to numerical instabilities. Spectral methods have been used successfully in many different fields in sciences and engineering because of their ability to give accurate solutions of differential equations. Khater et al.
The Chebyshev spectral collocation method has been used together with the fourth-order Runge-Kutta method to solve the nonlinear PDEs in this study. The Chebyshev spectral collocation is first applied to the NPDE and this yields a system of ordinary differential equations, which are solved using the fourth-order Runge-Kutta method.
Darvishi et al. Jacobs and Harley [ 31 ] and Tohidi and Kilicman [ 32 ] used spectral collocation directly for solving linear partial differential equations. Accuracy will be compromised if they implement their approach in solving nonlinear partial differential equations since they use Kronecker multiplication.
Chebyshev spectral methods are defined everywhere in the computational domain. Therefore, it is easy to get an accurate value of the function under consideration at any point of the domain, beside the collocation points. This property is often exploited, in particular to get a significant graphic representation of the solution, making the possible oscillations due to a wrong approximation of the derivative apparent.
Spectral collocation methods are easy to implement and are adaptable to various problems, including variable coefficient and nonlinear differential equations. The interest in using Chebyshev spectral methods in solving nonlinear PDEs stems from the fact that these methods require less grid points to achieve accurate results.
They are computational and efficient compared to traditional methods like finite difference and finite element methods. Chebyshev spectral collocation method has been used in conjunction with additional methods which may have their own drawbacks. Here, we provide an alternative method that is not dependent on another method to approximate the solution.
The main objective of this work is to introduce a new method that uses Chebyshev spectral collocation, bivariate Lagrange interpolation polynomials together with quasilinearisation techniques. The nonlinear evolution equations are first linearized using the quasilinearisation method. The Chebyshev spectral collocation method with Lagrange interpolation polynomials are applied independently in space and time variables of the linearized evolution partial differential equation.
We present the BI-SQLM algorithm in a general setting, where it can be used to solve any r th order nonlinear evolution equations. The results of the BI-SQLM are compared against known exact solutions that have been reported in the scientific literature. It is observed that the method achieves high accuracy with relatively fewer spatial grid points.
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It also converges fast to the exact solution and approximates the solution of the problem in a computationally efficient manner with simulations completed in fractions of a second in all cases. Tables are generated to show the order of accuracy of the method and time taken to compute the solutions. It is observed that, as the number of grid points is increased, the error decreases. Error graphs and graphs showing the excellent agreement of the exact and analytical solutions for all the nonlinear evolution equations are also presented.
The paper is organized as follows. The numerical simulations and results are presented in Section 4. Finally, we conclude in Section 5. Without loss of generality, we consider nonlinear PDEs of the form. The solution procedure assumes that the solution can be approximated by a bivariate Lagrange interpolation polynomial of the form.
The choice of the Chebyshev-Gauss-Lobatto grid points 5 ensures that there is a simple conversion of the continuous derivatives, in both space and time, to discrete derivatives at the grid points. The functions L i x are the characteristic Lagrange cardinal polynomials. The function L j t is defined in a similar manner. Before linearizing 3 , it is convenient to split H into its linear and nonlinear components and rewrite the governing equation in the form. We remark that this quasilinearization method QLM approach is a generalisation of the Newton-Raphson method and was first proposed by Bellman and Kalaba [ 33 ] for solving nonlinear boundary value problems.
Equation 9 can be expressed as. Substituting 10 into 8 , we get. Similarly, for an n th order derivative, we have. Substituting 16 into 12 we get.