A new spectral algorithm based on shifted second kind Chebyshev wavelets operational matrices of derivatives is introduced and used for solving linear and nonlinear secondorder twopoint boundary value problems. The main idea for obtaining spectral numerical solutions for these equations is essentially developed by reducing the linear or nonlinear equations with their initial and/or boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. Convergence analysis and some efficient specific illustrative examples including singular and Bratu type equations are considered to demonstrate the validity and the applicability of the method. Numerical results obtained are compared favorably with the analytical known solutions.
Spectral methods are one of the principal methods of discretization for the numerical solution of differential equations. The main advantage of these methods lies in their accuracy for a given number of unknowns (see, e.g., [
The subject of wavelets has recently drawn a great deal of attention from mathematical scientists in various disciplines. It is creating a common link between mathematicians, physicists, and electrical engineers. Wavelets theory is a relatively new and an emerging area in mathematical research. It has been applied to a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for wave form representation and segmentations, time frequency analysis, and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms (see [
The subject of nonlinear differential equations is a wellestablished part of mathematics, and its systematic development goes back to the early days of the development of calculus. Many recent advances in mathematics, paralleled by a renewed and flourishing interaction between mathematics, sciences, and engineering, have again shown that many phenomena in applied sciences, modelled by differential equations, will yield some mathematical explanation of these phenomena [
Evenorder differential equations have been extensively discussed by a large number of authors due to their great importance in various applications in many fields. For example, in the sequence of papers [
In this paper, we aim to give some algorithms for handling both linear and nonlinear secondorder boundary value problems based on introducing a new matrix of derivatives, then applying PetrovGalerkin method on linear equations and collocation method on nonlinear equations. High accurate spectral wavelets solutions are achieved with a small number of retained modes compared with the usual spectral methods, also we can handle singular differential equations with discontinuous variable coefficients.
Among the secondorder boundary value problems is the onedimensional Bratu problem which has a long history. Bratu’s own paper appeared in 1914 [
Simplification of the solid fuel ignition model in thermal combustion theory yields an elliptic nonlinear partial differential equation, namely, the Bratu problem. Also due to its use in a large variety of applications, many authors have contributed to the study of such problem. Some applications of Bratu problem are the model of thermal reaction process, the Chandrasekhar model of the expansion of the Universe, chemical reaction theory, nanotechnology, and radiative heat transfer (see [
The application of Legendre wavelets for solving differential and integral equations is thoroughly considered by many authors (see, [
In [
One approach for solving differential equations is based on converting the differential equations into integral equations through integration, approximating various signals involved in the equation by truncated orthogonal series, and using the operational matrix of integration, to eliminate the integral operations.
Special attentions have been given to applications of block pulse functions [
The main aim of this paper is to develop a new spectral algorithm for solving secondorder twopoint boundary value problems based on shifted second kind Chebyshev wavelets operational matrix of derivatives. The method reduces the differential equation with its initial and/or boundary conditions to a system of algebraic equations in the unknown expansion coefficients. Large systems of algebraic equations may lead to greater computational complexity and large storage requirements. However, the second kind Chebyshev wavelets are structurally sparse; this reduces drastically the computational complexity of solving the resulting algebraic system.
The structure of the paper is as follows. In Section
In the present section, we discuss some relevant properties of the second kind Chebyshev polynomials and their shifted forms.
It is well known that the second kind Chebyshev polynomials are defined on
The first derivative of second kind Chebyshev polynomials is given by
For a proof of Theorem
The shifted second kind Chebyshev polynomials are defined on
The first derivative of the shifted second kind Chebyshev polynomial is given by
Wavelets constitute of a family of functions constructed from dilation and translation of single function called the mother wavelet. When the dilation parameter
A function
We state and prove a theorem ascertaining that the second kind Chebyshev wavelet expansion of a function
A function
From (
A shifted second kind Chebyshev wavelets operational matrix of the first derivative is stated and proved in the following theorem.
Let
If we make use of the shifted second kind Chebyshev polynomials, then the
The operational matrix for the
In this section, we are interested in solving linear and nonlinear twopoint boundary value problems subject to homogenous or nonhomogenous initial or boundary conditions based on the wavelets operational matrices constructed in Section
Consider the linear secondorder differential equation
Consider the nonlinear differential equation
To find an approximate solution to
In this section, the presented algorithms in Section
Consider the following linear secondorder boundary value problem:
In Table
Maximum absolute error






















Consider the following linear secondorder boundary value problem:
In Table
Maximum absolute error






















Consider the following nonlinear initial value problem (see [
We solve (
Consider the following nonlinear secondorder boundary value problem:
In Table
Maximum absolute error










1 






Consider the following linear singular initial value problem (see [
Maximum absolute error





2 

6 

4 

8 

Comparison between different errors for Example






Legendre Wavelets [ 




SCWOMD 




Numerical and exact solutions of Example
Consider the following Bratu equation (see [
Maximum absolute error








10 






12 




14 




16 




18 




20 



Comparison between the best errors for Example
SCWOMD  Error in [ 
Error in [ 
Error in [ 
Error in [ 






Different solutions of Example
It is worth noting here that the obtained numerical results in the previous six solved examples are very accurate, although the number of retained modes in the spectral expansion is very few, and again the numerical results are compared favorably with the known analytical solutions.
In this paper, an algorithm for obtaining a numerical spectral solution for secondorder linear and nonlinear boundary value problems is discussed. The derivation of this algorithm is essentially based on constructing the shifted second kind Chebyshev wavelets operational matrix of differentiation. One of the main advantages of the presented algorithm is its availability for application on both linear and nonlinear secondorder boundary value problems including some singular equations and also Bratu type equations. Another advantage of the developed algorithm is that high accurate approximate solutions are achieved using a small number of the second kind Chebyshev wavelets. The obtained numerical results are compared favorably with the analytical ones.
This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support. Also, they would like to thank the referee for his valuable comments and suggestions which improved the paper into its present form.