Numerical solution of third order linear differential equations using generalized one-shot operational matrices in orthogonal hybrid function domain
Introduction
The main tool for tackling differential equations in the modern age is the numerical analysis, and to be explicit, numerical integration. Differential equations, in general, have a wide range of varieties [1], [2] along with different degrees of difficulties. For handling differential equations arising out of modern complex systems, numerical analysis is the forerunner of all solution techniques and modern day algorithms and number crunching capability of computers help in solving varieties of such equations to obtain practical solutions avoiding numerical instability. Work by Butcher [3] gives an exhaustive overview of numerical methods for solving ordinary differential equations. The 4th order Runge–Kutta method has undergone many improvements and modifications discussed by Butcher [2].
Differential equations having oscillatory solutions need special techniques for obtaining reasonable solution within tolerable error limits. Simos’s [4] work on modified Runge–Kutta methods for the numerical solution of ODEs with oscillating solutions tackles simultaneous first order ODE’s to obtain the required solution. Any method based upon numerical techniques for solving such equations is of interest in modern control theory and applications.
For more than three decades, solution of differential equations as well as integral equations was also attempted by employing piecewise constant basis functions (PCBF) like Walsh functions, block pulse functions etc. In such attempts, function approximation plays a pivotal role because, the initial error in function approximation is propagated in a cumulative manner at different stages of computations. Apart from orthogonal functions, orthogonal polynomials have also played their important role [5], [6] in this area.
In the present work, we have attempted to employ a new set of orthogonal hybrid functions (HF) for function approximation, integration of functions and finally solving differential equations.
In 1910, Haar functions [7] appeared as the first set of the PCBF family. This function set was entirely different from the ‘orthodox’ sine-cosine functions and was the genesis of a new class of orthogonal functions. Of this class, the block pulse function (BPF) [8], [9] set and its variants [10] proved to be the most efficient because of its simplicity and versatility in analysis [11] as well as synthesis [10], [12] of control systems. However, Navascqués [13] constructed a set of fractal functions close to the Haar [7] set for the approximation of discontinuities.
In 1998, an orthogonal set of sample-and-hold functions [14] were introduced by Deb et al and the same was applied to solve problems related to discrete time systems with zero order hold. The set of sample-and-hold functions approximate any square integrable function of Lebesgue measure in a piecewise constant manner and was proved to be more convenient for solving problems associated to sample-and-hold systems.
In 2003, orthogonal triangular functions [15], [16], [17] were introduced by Deb et al. and the same were applied to control system related problems including analysis and system identification. The set of triangular functions approximate any square integrable function in a piecewise linear manner.
In this paper, a set of orthogonal hybrid functions (HF) [18], [19], which is a combination of sample-and-hold function and triangular function, is presented. However, it may be noted that several works have been published in the past where the term ‘hybrid’ function was used. For example, Haddadi et al. [20] discussed about the properties of functions named ‘hybrid’ functions, consisting of block-pulse functions and Bernoulli polynomials.
We utilize the ‘new’ hybrid function set for
- (i)
Function approximation and comparison of its accuracy with the exact solution and approximations via block pulse functions and Legendre polynomials [21], [22],
- (ii)
Computation of the operational matrices for integration in hybrid function (HF) domain,
- (iii)
Integration of time functions using the operational matrices for integration,
- (iv)
Computation of generalized one-shot operational matrices for multiple integration,
- (v)
Multiple integration of time functions using one-shot operational matrices,
- (vi)
Solution of linear third order differential equations using one-shot operational matrices for integration.
The work of Ref. [23] has resulted in symbolic algorithms for linear boundary problems, which can be seen as a further step in the algorithmization of analysis by algebraization and as a contribution to the emerging field of symbolic analysis—dealing with problems from analysis, notably differential equations.
Solutions for a class of degenerate, nonlinear, nonlocal boundary value problems were discussed by Akyildiz et al. [24] where as Ramos [25] focuses upon analytical solutions to autonomous, nonlinear, third-order nonlinear ordinary differential equations invariant under time and space reversals. These solutions are obtained by means of transformations and include periodic as well as non-periodic behavior.
Mathematical studies of third-order nonlinear ordinary differential equations include proof of the stability and boundedness of solutions of nonlinear vector differential equations by means of Lyapunov’s second method. Mathematical modeling of several physical phenomena sometimes result in third-order nonlinear ordinary differential equations, e.g., the current in a vacuum tube circuit where the nonlinearities arise from the nonlinear characteristics of the tube. Third-order ordinary differential equations also model the dynamics of nuclear spin generators, thermo-mechanical oscillators in fluids, transverse motions of piano strings, interactions between an elastic sphere and a surrounding fluid, vibrations of a mass attached to two horizontal strings and subject to aerodynamic forces, control systems, etc.
All these work emphasizes the importance of third order differential equations in the area of physics and engineering.
For determining the solution of a third order differential equation by this new approach, all the necessary computations are executed using MATLAB [26]. Results thus obtained for the cases (i) and (iii) above are compared with the exact solutions. The results obtained for item no. (v) are compared with the results computed via exact analysis.
Section snippets
Block pulse functions (BPF) [8]
The ith member of a set of well-known BPF, Ψ(m)(t), in [0, T) comprised of m component functions, is defined asWhere, and i = 0, 1, 2,…, (m − 1).
Fig. 1 shows a set of eight block pulse functions.
The BPF set is orthogonal as well as complete in .
A square integrable time function f(t) of Lebesgue measure may be expanded [8], [9] into an m-term BPF series in with minimum mean integral square error (MISE) [8], as
Hybrid function (HF): a combination of SHF and TF [18,19]
We can use a set of sample-and-hold functions and the RHTF set of triangular functions to form a hybrid function set, which we name a ‘Hybrid function set’. To define a hybrid function (HF) set, we express the ith member Hi(t) of the m-set hybrid function H(m)(t) in 0 ⩽ t < T aswhere, i = 0, 1, 2, …, (m − 1), ai and bi are scaling constants. For convenience, in the following, we write T instead of T2.
While the block pulse function set provides us with a staircase solution, the
Method of function approximation by hybrid functions
Consider a function f(t) in an interval . If we take (m + 1) equidistant samples of the function with a sampling period h (i.e., T = mh), we can write
It is observed from Fig. 4 that hybrid function method of function approximation mainly depends upon samples of the involved function. This is a pretty strong advantage which outweighs the traditional approach of function approximation
Operational matrices for integration
A hybrid function set is a combination of a sample-and-hold function set and a triangular function set (RHTF). In order to derive the operational matrices for integration in hybrid function domain, we proceed in a familiar manner [8], [27] adopted for Walsh and block pulse functions. Here, for hybrid function set, we consider both the component function sets separately.
Integration of functions using operational matrices
Let f(t) be a square integrable function which can be expanded in hybrid function domain from Eq. (12) aswhere, T denotes transpose.
Integrating Eq. (31) with respect to t, we get
Now we use (32) to perform integration of a few simple square integrable functions.
Repeated integration using 1st order integration matrices only
We know from Eqs. (20), (27) that
So, we can write
Using the relations (28), (30) in (36), we have
For n times repeated integration of the vector, we havewhere,
Similarly, for the vector T(m) we have
One-shot integration operational matrices for repeated integration
It is noted that the operation of first order integration using operational matrices P1ss, P1st, P1ts, P1tt, the result of integration is somewhat approximate. If we carry on repeated integration using these matrices, error will surely accumulate and higher order integrations in HF domain may become so corrupted that may lead to a fiasco.
For this reason, we present in the following more accurate one-shot operational matrices of higher orders suitable for computation of function integration with
Numerical example
Let us now take up an example to compare the efficiencies of repeated use of 1st order integration matrices and higher order one-shot integration matrices.
Consider the functionWhere, DS and DT are HF domain coefficient vectors of f(t) known from the actual samples of the function t.Where, CS and CT are HF domain coefficient vectors known from the actual samples of the function t.
Now we perform single, double and
Two theorems
It should be noted that all the operational matrices P, P1ss, P1st, P1ts, P1tt, P2ss, P2st, P2ts, P2tt, P3ss, P3st, P3ts, P3tt, …, Pnss, Pnst, Pnts, Pntt, … are of regular upper triangular nature and may be represented by S having the following general form:where, the delay matrix Q [12] is given by
We present the following two theorems regarding commutative property of matrices of class S and its polynomials. Theorem 1 If a regular upper triangular matrix S of order m can be
Solution of third order differential equation
We present two methods in the following based upon
- (i)
The repeated use of first order integration matrices.
- (ii)
The use of higher order one-shot integration matrices.
Conclusion
In this paper, a new set of orthogonal functions, comprised of triangular functions and sample-and-hold functions, termed hybrid functions (HF), has been proposed. This new function set prove to be efficient for function approximation which is established via one illustrative example. Function approximation via Legendre polynomials and block pulse functions are also discussed supported by figures and tables.
As in case of Walsh and block pulse functions, the operational matrices for integration,
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