We consider a , rst-order partial di, erential equation with constant irreversible co-e, cients in a Banach space in the regular case. The equation is split into equations in subspaces, in which non-degenerate subsystems are obtained. We obtain an ana-lytical solution of each system with Showalter-type conditions. Finally, an example is given to illustrate the theoretical results.

In the present paper, we obtain the traveling wave fuzzy solution for the fuzzy linear Transport equation and the fuzzy Wave equation by considering the type of generalized Hukuhara di erentiability. The d'Alembert's formulas for the fuzzy Wave equation obtained by Considering the type of gHdi erentiability of the solution. Also, The existence and the uniqueness of these solutions and the stability of the fuzzy Wave equation are shown. Furthermore, Some examples are solved to illustrate the technique.

One of the aims of this article is to investigate the solvability and unsolvability conditions for fractional cohomological equation , f = g, on Tn. We prove that if f is not analytic, then fractional integro-di, erential equation I1􀀀, ,t D,x u(x,t) + iI1􀀀, ,x D,t u(x,t) = f(t) has no solution in C1(B) with 0 < ,,1. We also obtain solutions for the space-time fractional heat equations on S1 and Tn. At the end of this article, there are examples of fractional partial di, erential equations and a fractional integral equation together with their solutions.

In the present article, we utilize the Conformable Double Laplace Transform Method (CDLTM) to get the exact solutions of a wide class of Conformable fractional di, er-ential in mathematical physics. The results obtained show that the proposed method is e, cient, reliable and easy to be implemented on related linear problems in applied mathematics and physics. Moreover, the (CDLTM) has a small computational size as compared to other methods.

The purpose of this paper is to present a numerical scheme for solving time-fractional partial di, erential equation based on cubic B-spline quasi-interpolation. For this pur-pose, , rst we will approximate the time-fractional derivative by Laplace transform method and then by using of cubic B-spline quasi-interpolation, the spatial deriva-tives are approximated. Moreover, the stability of this method is studied. Finally, European call and put options are priced and we will show that the results are good agreement with the other methods. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement.

In this research paper, a numerical method is presented for solving second-order hybrid fuzzy differential equations by using fuzzy Taylor expansion under generalized Hukuhara di erentiability and also with convergence theorem. Also, the method is illustrated by solving several numerical examples. The  nal results showed that the solution of the second-order hybrid fuzzy di erential equations.

In this paper, an e, cient numerical method is used to provide the approximate so-lution of distributed-order fractional partial di, erential equations (DFPDEs). The proposed method is based on the fractional integral operator of fractional-order Bernoulli-Legendre functions and the collocation scheme. In our technique, by ap-proximating functions that appear in the DFPDEs by fractional-order Bernoulli func-tions in space and fractional-order Legendre functions in time using Gauss-Legendre numerical integration, the under study problem is converted to a system of algebraic equations. This system is solved by using Newton's iterative scheme, and the nu-merical solution of DFPDEs is obtained. Finally, some numerical experiments are included to show the accuracy, e, ciency, and applicability of the proposed method.

This paper deals with the numerical treatment of singularly perturbed parabolic reaction-di, usion initial boundary value problems. Introducing a , tting parameter into the asymptotic solution and applying average , nite di, erence approximation, a , tted operator , nite di, erence method is developed for solving the problem. To ac-celerate the rate of convergence of the method, Richardson extrapolation technique is applied. The consistency and stability of the proposed method have been established very well to ensure the convergence of the method. Numerical experimentation is carried out on some model problems and both the results are presented in tables and graphs. The numerical results are compared with , ndings of some methods existing in the literature and found to be more accurate. Generally, the formulated method is consistent, stable, and more accurate than some methods existing in the literature for solving singularly perturbed parabolic reaction-di, usion initial boundary value problems.

In this paper a  nite di erence method for solving 2-dimensional di usion equation is presented. The method employs Crank-Nicolson scheme to improve -nite di erence formulation and its convergence and sta-bility. The obtained solution will be a recursive formula in each step of which a system of linear equations should be solved. Given the speci c form of obtained matri-ces, rather than solving the problem in each step using conventional iterative methods, a closed-form solution is formulated.

In this paper, we consider some boundary value problems (BVP) for fractional order partial di erential equations (FPDE) with non-local boundary conditions. The solutions of these problems are pre-sented as series solutions analytically via modi ed Mittag-Le er func-tions. These functions have been modi ed by authors such that their derivatives are invariant with respect to fractional derivative. The pre-sented solutions for these problems are as in nite series. Convergence of series solutions and uniqueness of them are established by general theory of mathematical analysis and theory of ODEs.