In this article, a new homotopy technique is presented for the mathematical analysis of finding the solution of a first-order inhomogeneous partial differential equationThe homotopy perturbation method (HPM) and the decomposition of a source function are used together to develop this new technique. The homotopy constructed in this technique is based on the decomposition of a source function. Various decompositions of source functions lead to various homotopies. Using the fact that the decomposition of a source function affects the convergence of a solution leads us to development of a new method for the decomposition of a source function to accelerate the convergence of a solution. The purpose of this study is to show that constructing the proper homotopy by decomposing f (x, y) in a correct way determines the solution with less computational work than using the existing approach while supplying quantitatively reliable results. Moreover, this method can be generalized to all inhomogeneous PDE problems.
In this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(x) in the linear parabolic equation u t (x, t) = (k(x)u x (x, t)) x , with Dirichlet boundary conditions u(0, t) = ψ 0 , u(1, t) = ψ 1 . Main goal of this study is to investigate the distinguishability of the input-output mappingssemigroup theory. In this paper, we show that if the null space of the semigroup T (t) consists of only zero function, then the input-output mappings Φ[·] and Ψ [·] have the distinguishability property. Moreover, the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x = 0 and x = 1, respectively, can be determined explicitly by making use of measured output data (boundary observations) f (t) := k(0)u x (0, t) or/and h(t) := k(1)u x (1, t). In addition to these, the values k (0) and k (1) of the unknown coefficient k(x) at x = 0 and x = 1, respectively, are also determined via the input data. Furthermore, it is shown that measured output data f (t) and h(t) can be determined analytically, by an integral representation. Hence the input-output mappings Φ[·] : K → C 1 [0, T ], Ψ [·] : K → C 1 [0, T ] are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient k(x) at the end points x = 0 and x = 1.
The study in this paper mainly concerns the inverse problem of determining an unknown source function in the linear fractional differential equation with variable coefficient using Adomian decomposition method (ADM). We apply ADM to determine the continuous right hand side functionsfxandftin the heat-like diffusion equationsDtαux,t=hxuxxx,t+fxandDtαux,t=hxuxxx,t+ft, respectively. The results reveal that ADM is very effective and simple for the inverse problem of determining the source function.
In this study, we deal with the problem of constructing semianalytical solution of mathematical problems including space-time-fractional linear and nonlinear differential equations. The method, called Shehu Variational Iteration Method (SVIM), applied in this study is a combination of Shehu transform (ST) and variational iteration method (VIM). First, ST is utilized to reduce the time-fractional differential equation with fractional derivative in Liouville-Caputo sense into an integer-order differential equation. Later, VIM is implemented to construct the solution of reduced differential equation. The convergence analysis of this method and illustrated examples confirm that the proposed method is one of best procedures to tackle space-time-fractional differential equations.
The motivation of this study is to determine the analytic solution of initial boundary value problem including time fractional differential equation with Neumann boundary conditions in one dimension. By making use of seperation of variables, the solution is constructed in the form of a Fourier series with respect to the eigenfunctions of a corresponding Sturm-Liouville eigenvalue problem.
We develop a formulation for the analytic or approximate solution of fractional differential equations (FDEs) by using respectively the analytic or approximate solution of the differential equation, obtained by making fractional order of the original problem integer order. It is shown that this method works for FDEs very well. The results reveal that it is very effective and simple in determination of solutions of FDEs.
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