In this work, we developed a new numerical method based on shifted Jacobian polynomials for solving linear and nonlinear initial value problem and boundary value problem of partial differential equation. These methods are called the Multistep Shift Jacobi Tau (M-SJT) and the Multistep Shift Jacobi Collocation (M-SJC) method, respectively. 9 3.1.2 The Jacobi Tau (M-SJT) multistep lag method for the initial value problem 11 3.1.3 Numerical results of linear FDE.
29 4.3 Multistage Jacobi Displacement (M-SJC) method for the boundary value problem 30 4.3.1 Shot method for the boundary value problem. In this work, we focus on shifted Jacobi polynomials for solving linear and nonlinear fractional problems. Thus, for the given nonlinear problem, the Shifted Jacobi Collocation (SJC) method is introduced to find an exact approximation.
We also use the methods based on the shifted Jacobi polynomials for solving fractional boundary value problems. In Chapter 3, we concentrate on methods based on shifted Jacobi polynomials for solving initial value problems of fractional differential equations, such as the Shifted Jacobi Collocation method and the Shifted Jacobi Tau method. In Chapter 4, we present Shifted Jacobi Collocation for solving the boundary value problem of the fractional differential equation.
Since the main tool for solving initial and boundary value problems of the fractional differential equation is based on shifted Jacobi polynomials, their definitions are presented as well as some properties.
Fractional Calculus
Having established these fundamental properties of the Riemann-Liouville integral operator, we now turn to the corresponding differential operators[4]. It turns out that the Riemann-Liouville derivatives have certain disadvantages when modeling real-world phenomena with fractional differential equations. Assume that n≥0 and that f is such that Then[f−Tm−1[f;a]] exists, where Tm−1[f;a] denotes the Taylor polynomial of degree m−1 for the function f, centered at a and m=dne.
This essentially states that the eigenfunction of Caputo differential operators can be expressed in terms of Mittag-Leffler functions. Here we used the fact that, in view of the convergence properties of the series defining the Mittage-Leffler function, we can interchange first summation and differentiation and later summation and integration. It is often important to have knowledge about the asymptotic behavior of the Mittag-Leffler functions.
Shifted Jacobi Polynomials
Now suppose that the function uΩ(x) can be approximated using (N + 1) terms of shifted Jacobi polynomials. Of these Shifted Jacobi polynomials, the most commonly used Shifted Jacobi polynomials in numerical analysis are the shifted Legendre polynomialsPΩ,n; and the shifted Chebyshev polynomials of the first kindTΩ,n.
Model Problem
It is also clear that the matrix system becomes larger if the solution is approximated with much shifted Jacobi polynomials. In this section, we propose an efficient computational method, namely the Multistage Shifted Jacobi Tau(M-SJT) method. The basic idea of the M-SJT method is to apply the standard SJT method to the problem in each subdomain.
In this section, we demonstrate numerical results of initial value problem using the M-SJT method. If ν is close to 1, we can obtain a more accurate result due to less proportion of the memory term. We can know the maximum errors and the convergence rate of N(degree) and the convergence rate of n(step size) for the changes in each of n and N.
Also, the convergence rate of n (step size) is about 1.5 and the convergence rate of N (degree) is about 3. We can know the maximum errors and convergence rate of n (step size) for the changes in each of the n and N.
Model Problem
It is also clear that the equations become larger if the solution is approximated by many shifted Jacobi polynomials. In this section, we propose an efficient computational method, namely the Multistage Shifted Jacobi Collocation (M-SJC) method. The basic idea of the M-SJC method is to apply the standard SJC method to the problem in each sub-domain.
In this section, we demonstrate numerical results of initial value problem using the M-SJC method. Also the rate of convergence for n (step size) and the rate of convergence for N (degree) depend on ν. Also, the convergence rate for n (step size) is approximately 2.3 and the convergence rate for N(degrees) is approximately 4.
Model Problem
Shifted Jacobi collocation (SJC) method for boundary value problem
Multistage Shifted Jacobi collocation (M-SJC) method for boundary value problem
We can clearly obtain from ∂f∂u and ∂D∂fνu, therefore problem (4.3.3) can be solved by fractional order methods as illustrated above.
Numerical results of Fractional Boundary Value Differential Equation
We want to converge red value because the exact initial value of s is zero in the Example 5. Considering the sensitivity in the Table, ex5 was solved using M-SJC. α= 0 and β = 0) for Example 5. In this work we developed a new numerical method based on the shifted Jacobi polynomials for solving linear and non-linear initial value problem and boundary value problem of fractional differential equation.
Using a multi-step methodology, we extend conventional spectral approaches such as the shifted Jacobi Tau method for the linear problem and the shifted Jacobi collocation method for the nonlinear problem. In other words, on the equally spaced partition P : 0 = x0 < x1 < · · · < xn = 1, the conventional spectral methods called multilevel shifted Jacobi Tau(M-SJT) and multilevel shifted Jacobi collocation method are used in each subdomain (M-SJC). From several illustrative examples, it is concluded that the M-SJT method has better accuracy than the standard SJT for the linear initial value problem and that the M-SJC method provides a more accurate approximation than the standard SJC method for the nonlinear initial value problem.
Especially for the less regularization of the solution, such as Example 4, the standard method gives a poor approximation due to smooth polynomial basis functions, whereas the proposed method achieves more accurate approximation because the polynomial basis functions can be applied to the locally smooth subdomain. In addition, we extend the proposed methods for solving nonlinear boundary value problem for fractional differential equations. Since all proposed methods are developed to solve the initial problem, it is necessary to convert the boundary value problem to the original problem.
From the numerical example, it is clear that M-SJC has a better accuracy than standard SJC for the nonlinear boundary value problem. It is easy to see that the matrix system according to the standard SJT becomes larger if many jacobi polynomials are applied as basis functions. However, in the multistage approach, the size of matrix system can be reduced because the method is applied to the prob-
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ACKNOWLEDGEMENTS
