A Fractional Lie Group Method For Anomalous Diffusion Equations

Lie group method provides an efficient tool to solve a differential equation. This paper suggests a fractional partner for fractional partial differential equations using a fractional characteristic method. A space-time fractional diffusion equation is used as an example to illustrate the effectiveness of the Lie group method.Comment: 5 pages,in pres

Fractional Variational Iteration Method for Fractional Nonlinear Differential Equations

Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to overcome such shortcomings, a fractional variational iteration method is proposed. The Lagrange multipliers can be identified explicitly based on fractional variational theory.Comment: 12 pages, 1 figure

Fractional variational iteration method via modified Riemann–Liouville derivative

AbstractThe aim of this paper is to present an efficient and reliable treatment of the variational iteration method (VIM) for partial differential equations with fractional time derivative. The fractional derivative is described in the Jumarie sense. The obtained results are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, efficient and easy to implement

Numerical Solution of Nonlinear Fractional Volterra Integro-Differential Equations via Bernoulli Polynomials

This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main problems to the solution of algebraic equations systems by expanding the required approximate solutions as the linear combination of the Bernoulli polynomials. Several examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method

Penerapan Metode Iterasi Variasional Untuk Mencari Solusi Numerik Pada Persamaan Fokker-Planck

Persamaan Fokker-Planck merupakan persamaan diferensial parsial yang penting dalam pemodelan fenomena stokastik. Dalam upaya untuk menyelesaikan persamaan ini, berbagai metode numerik telah dikembangkan, termasuk metode iterasi variasional dan metode garis. Metode iterasi variasional melibatkan pendekatan iteratif yang menggabungkan komponen waktu dan spasial untuk menyelesaikan persamaan Fokker-Planck. Kriteria penghentian berdasarkan nilai galat maksimum dan galat rata-rata digunakan untuk memastikan konvergensi solusi. Di sisi lain, metode garis menggunakan skema finite difference dalam pendekatan diskritisasi ruang dan waktu untuk menyelesaikan persamaan tersebut. Implementasi kedua metode dilakukan menggunakan bahasa pemrograman Python. Beberapa kasus uji digunakan untuk menguji performa dan akurasi kedua metode, termasuk kasus dengan koefisien difusi dan potensial konstan, koefisien difusi dan potensial yang bervariasi secara linear, serta koefisien difusi dan potensial yang bervariasi secara eksponensial. Hasil pengujian menunjukkan bahwa kedua metode dapat menghasilkan solusi numerik yang konvergen. Namun, terdapat perbedaan dalam akurasi dan efisiensi antara kedua metode. Metode iterasi variasional cenderung memberikan solusi yang lebih akurat, namun memerlukan waktu komputasi yang lebih lama. Di sisi lain, metode garis menunjukkan efisiensi yang lebih tinggi, tetapi dengan tingkat akurasi yang lebih kasar. Dalam kesimpulannya, penelitian ini memberikan pemahaman yang lebih baik tentang penerapan metode numerik dalam menyelesaikan persamaan Fokker-Planck pada model dinamika sistem stokastik. Hasil ini dapat menjadi landasan untuk pengembangan lebih lanjut dalam pemodelan sistem stokastik dan pemahaman fenomena yang terkait. Dengan memahami kelebihan dan kekurangan masing-masing metode, penelitian ini memberikan wawasan tentang pemilihan metode yang sesuai tergantung pada kebutuhan akurasi dan efisiensi yang diinginkan

Optimal variational iteration method for parametric boundary value problem

Mathematical applications in engineering have a long history. One of the most well-known analytical techniques, the optimal variational iteration method (OVIM), is utilized to construct a quick and accurate algorithm for a special fourth-order ordinary initial value problem. Many researchers have discussed the problem involving a parameter c. We solve the parametric boundary value problem that can't be addressed using conventional analytical methods for greater values of c using a new method and a convergence control parameter h. We achieve a convergent solution no matter how huge c is. For the approximation of the convergence control parameter h, two strategies have been discussed. The advantages of one technique over another have been demonstrated. Optimal variational iteration method can be seen as an effective technique to solve parametric boundary value problem

Homotopy decomposition method for solving higher-order time-fractional diffusion equation via modified beta derivative

In this paper, the homotopy decomposition method with a modified definition of beta fractional derivative is adopted to find approximate solutions of higher-dimensional time-fractional diffusion equations. To apply this method, we find the modified beta integral for both sides of a fractional differential equation first, then using homotopy decomposition method we can obtain the solution of the integral equation in a series form. We compare the solutions obtained by the proposed method with the exact solutions obtained using fractional variational homotopy perturbation iteration method via modified Riemann-Liouville derivative. The comparison shows that the results are in a good agreement

Analysis of Fractal Wave Equations by Local Fractional Fourier Series Method

The fractal wave equations with local fractional derivatives are investigated in this paper. The analytical solutions are obtained by using local fractional Fourier series method. The present method is very efficient and accurate to process a class of local fractional differential equations