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It is well-known that one-dimensional time fractional diffusion-wave equations with variable coefficients can be reduced to ordinary fractional differential equations and systems of linear fractional differential equations via scaling…
For the Allen-Cahn equation, it is well known that there is a monotone standing wave joining with the balanced wells of the potential. In this paper we study the existence of traveling wave solutions for the Allen-Cahn equation on an…
A new rigorous approach for precise and efficient calculation of light propagation along non-uniform waveguides is presented. Resonant states of a uniform waveguide, which satisfy outgoing-wave boundary conditions, form a natural basis for…
Often a non-linear mechanical problem is formulated as a non-linear differential equation. A new method is introduced to find out new solutions of non-linear differential equations if one of the solutions of a given non-linear differential…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…
There are many methods for finding a particular solution to a nonhomogeneous linear ordinary differential equation (ODE) with constant coefficients. The method of undetermined coefficients, Laplace transform method and differential operator…
A set of traveling wave solution to convection-reaction-diffusion equation is studied by means of methods of local nonlinear analysis and numerical simulation. It is shown the existence of compactly supported solutions as well as solitary…
A method is given for finding roots of a one-variable function using Taylor's expansion of that function and fractional derivative calculated at a suitable tangent point without using Newton's method, but is regarded as a variant of Halley…
A class of particular travelling wave solutions of the generalized Benjamin-Bona-Mahony equation is studied systematically using the factorization technique. Then, the general travelling wave solutions of Benjamin-Bona-Mahony equation, and…
The Fractional Diffusion Equation (FDE) is a mathematical model that describes anomalous transport phenomena characterized by non-local and long-range dependencies which deviate from the traditional behavior of diffusion. Solving this…
A method is presented for calculating solutions to differential equations analytically for a variety of problems in physics. An iteration procedure based on the recently proposed BLUES (Beyond Linear Use of Equation Superposition) function…
Exact solutions of the Kawahara equation by Assas [L.M.B. Assas, J. Com. Appl. Math. 233 (2009) 97--102] are analyzed. It is shown that all solutions do not satisfy the Kawahara equation and consequently all nontrivial solutions by Assas…
In this paper, we study the global existence and regularity of H\"older continuous solutions for a series of nonlinear partial differential equations describing nonlinear waves.
We use a simple method that leads to the integrals involved in obtaining the traveling wave solutions of wave equations with one and two exponential nonlinearities. When the constant term in the integrand is zero, implicit solutions in…
In this paper, we acquire the soliton solutions of the nonlinear Schrodinger's equation with dual power-law nonlinearity. Primiraly, we use the extended trial equation method to find exact solutions of this equation. Then, we attain some…
An algorithm for numerically computing the exponential of a matrix is presented. We have derived a polynomial expansion of $e^x$ by computing it as an initial value problem using a symbolic programming language. This algorithm is shown to…
In this article a new approach in solving time fractional partial differential equations is introduced, that is, the ARA-residual power series method. The main idea of this technique, depends on applying the ARA-transform and using Taylor's…
The functional flow equations for the Legendre effective action, with respect to changes in a smooth cutoff, are approximated by a derivative expansion; no other approximation is made. This results in a set of coupled non-linear…
By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before…
The Lane-Emden type equations are employed in the modelling of several phenomena in the areas of mathematical physics and astrophysics . In this paper a new numerical method is applied to investigate some well-known classes of Lane-Emden…