This paper proposed a new type of numerical scheme for solving multi-term fractional nonlinear differential equation with two point boundary value problems (FBVP). In new techniques, FBVP is transposed to System of Fractional Nonlinear Initial Value Problems (FIVP). To solve the FIVP system, the Higher-Order Predictor-Corrector Method [1](HOM) is used.
Furthermore, we use a recording method based on Newton and Halley's methods to approximate the unknown initial values of the system. Several numerical experiments show that the proposed methods achieve the same convergence rate in the HOM. Keywords: Capuoto fractional derivative, fractional differential equation, fractional order boundary value problem, predictor-corrector methods, nonlinear recording methods.
Model Problem
Preliminaries
Moreover, it is an immediate consequence of the fundamental theorem that the following relation holds for the operators Then and Yes. Now, the following definition seems quite natural. for a≤t≤b, is called the Riemann-Liouville fractional integral operator of order α. But it turns out that the Riemann-Liouville derivative has some drawbacks when trying to model real phenomena with fractional differential equations.
Then, there exists a number b such that |a−b|=h and there also exists the function y ∈C[a, b] that solves the initial value problem (I.16). Then, the next Lemma will show which equation is equivalent to the solution of (I.16) and that the following lemma will fit the numerical method for . The function∈C[a,b]is a solution of the initial value problem (I.16) if and only if it is a solution of the nonlinear Volterra integral equation of the second kind.
Define the set G as in theorem (1.2.5) and let the function f :G→ R continue and satisfy a Lipschitz condition with respect to the second variable, i.e., then the numerical method for needs less cost computational than the conventional one. one for the fractional boundary value problem.
Description about Fractional Boundary Value Problem
If conventional method is used to solve the above problem, solving a dense matrix and multidimensional solver requires a lot of computational cost. And that recording methods are repeated when the error between resulting data is reduced. will show the outline of the above explanation. Whenever the function∈Cdαn[a, b]is a solution of the polynomial equation II.2 with initial conditions, the vector-valued function Y := (y1,· · · , yn)T will med.
When the vector-valued function Y := (y1,· · ·, yn)T is a solution of the multi-order fractional differential system (II.4a) with initial condition (II.4b), the function y := y1 is a solution of the multiterm equation II.2 with initial conditions.
Numerical Methods
The first one is “Predictor-Evaluate Corrector-Evaluate Method(PECE)” type [14], and another is “Second-order High-order Predictor Corrector Method(SHOM)” type [1]. Suppose f(·, y(·))∈C2[a, b]and furthermore is Lipschitz continuous in the second argument in Theorem (1.2.6) then we have. II.18) Therefore, SHOM has a better convergence rate than PECE. Global error of third order HOM Suppose(·, y(·))∈C3[a, b]and is Lipschitz continuous in the second argument in Theorem (1.2.6) then we have.
And error functionF(s) is defined by,. II.29) Then, to find zero of error function F(s), Newton's and Halley's methods are used. Through that process, the amount of error will show, and if it is greater, Newton's Method will be repeated enough when F(s) becomes almost zero. So, first predictor yn+1P and wPn+1 are found by PECE or SHOM or THOM (in this paper SHOM and THOM are used), then zn+1 can be calculated.
Second, the detected zn+1 is used as a predictor to find wn+1, and finally yn+1 can be found using wn+1 as a predictor. Also the above structure can be repeated, because if a circular is realized then there are areyn+1 and wn+1.
Numerical Results
And the convergence rate for HOMs, HMs and IHMs are similar, but the estimated error is different.
Numerical Results
Through a similar process, the above variable order two-point boundary value problem can be transformed into a variable order system of initial value problem, and it can also use Newton and Halley's shooting method in a similar way. But that matter is not considered in this paper, so it will probably be future works.
Variable order Fractional Two-Point Boundary Value Problem
But that case is not considered in this paper, so it will probably be future works. a) SN Variable order with Dirichlet boundary condition. But that of PECE is affected by the fractional order and is worse than the previous three methods. System of FIVP, which is transposed from FBVP, has a fraction term and it can be omitted.
The numerical results of FBVP with the first-order derivative term show that the maximum error between the exact value and the approximate solution becomes smaller when α1 approaches 1. Finally, although there is no theoretical consideration and there is only the intuitive idea from the theoretical content of fixed order, random numerical examples of variable order are included. Thus, that case must need precise theories, because those numerical results might just be a coincidence.
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