A Novel Analytical Approach for the Solution of Fractional-Order Diffusion-Wave Equations (original) (raw)
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Journal of Function Spaces
In general, solving fractional partial differential equations either numerically or analytically is a difficult task. However, mathematicians have tried their best to make the task easy and promoted various techniques for their solutions. In this regard, a very prominent and accurate technique, which is known as the new technique of the Adomian decomposition method, is developed and presented for the solution of the initial-boundary value problem of the diffusion equation with fractional view analysis. The suggested model is an important mathematical model to study the behavior of degrees of memory in diffusing materials. Some important results for the given model at different fractional orders of the derivatives are achieved. Graphs show the obtained results to confirm the accuracy and validity of the suggested technique. These results are in good contact with the physical dynamics of the targeted problems. The obtained results for both fractional and integer orders problems are ex...
IEEE Access
A new technique of the Adomian decomposition method is developed and applied in this research article to solve two-term diffusion wave and fractional telegraph equations with initial-boundary conditions. The proposed technique is used to solve problems of both fractional and integer orders of the telegraph equations. The fractional-order solutions provide useful information about the data transmission from one point to another. The solutions are obtained in the form of an infinite series, demonstrating a high rate of accuracy from fractional to integer orders of the problems. The technique's accuracy is verified by drawing various fractional and integer order plots and tables. The fractional-order plots demonstrate that the solutions have a higher rate of accuracy, and the different dynamical behaviors of the problems are revealed as a result. It is discovered that the new Adomian decomposition method is the best option for solving initial-boundary value problems. The new approximations of each solution improve the method's accuracy. As a result, it is suggested that the method can be applied to other problems with initial-boundary conditions. INDEX TERMS Adomian decomposition method; initial-boundary value problems; Caputo derivative; two-term diffusion-wave equations; time-fractional telegraph equations.
Solution of Fractional Heat-Like and Fractional Wave-Like Equation by Using Modern Strategy
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This paper introduces a novel form of the Adomian decomposition (ADM) method for solving fractional-order heat-like and wave-like equations with starting and boundary value problems. The derivations are provided in the sense of Caputo. In order to help understanding, the generalised formulation of the current approach is provided. Several numerical examples of fractional-order diffusion-wave equations (FDWEs) are solved using the suggested method in this context. In addition to examining the applicability of the suggested method to the solving of fractional-order heat-like and wave-like equations, a graphical depiction of the solutions to three instructive cases was constructed. Solution graphs were arrived at for integer and fractional-order problems. The derived and exact solutions to integer-order problems were found to be in excellent agreement. The subject of the present research endeavour is the convergence of fractional-order solutions. This strategy is considered to be the m...
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The Adomian decomposition method is a semi-analytical method for solving ordinary and partial nonlinear differential equations. The aim of this paper is to apply Adomian decomposition method to obtain approximate solutions of nonlinear fractional order partial differential equations with fractional derivatives. The fractional derivatives are taken in the sense of Caputo. The solutions of fractional PDEs are calculated in the form of convergent series. Approximate solutions obtained through the decomposition method have been numerically evaluated, and presented in the form of graphs and tables, and then these solutions are compared with the exact solutions and the results rendering the explicitness, effectiveness and good accuracy of the applied method. Finally, it is observed that the applied method (i.e. Adomian decomposition method) is prevailing and convergent method for the solutions of nonlinear fractional-order partial deferential problems.
Delta University Scientific Journal, 2023
In this paper, we apply the Adomian decomposition method (ADM) for solving Fractional Differential Equations (FDEs) with some modifications to the traditional method. The aim of this paper is to make ADM more efficient, rapid in convergence, and easy to use, so we will discuss two modifications. We use the reliable modification to simplify calculations. For difficulties in symbolic integration, we use a numerical implementation method. All these modifications were applied to the integer-order case, so we would apply it to FDEs. Some numerical results are given from solving these cases and comparing the solution with the ADM method.
2013
Fractional calculus has been used in many areas of sciences and technologies. This is the consequences of the elementary calculus. The order of the derivative in elementary calculus is integer, n. The nth derivative was changed to a for fractional calculus, where a is a fraction number or complex number. Fractional diffusion equation is one of the examples of fractional derivative equation. This study will focus on the solving fractional diffusion equation using variational iteration method and Adomian decomposition method to obtain an approximate solution to the fractional differential equation. Graphical output may explain further the results obtained. In certain problems the use of fractional differential equation gives more accurate representation rather than using elementary differential equation. Adomian decomposition method is easier in solving fractional diffusion equation since there is no nonlinear term in the equation. However, variational iteration method is more suitabl...