The homotopy analysis method (HAM) is applied to solve linear and nonlinear fractional partial differential equations (fPDEs). The fractional derivatives are described by Caputo's sense. Series solutions of the fPDEs are obtained. A convergence theorem for the series solution is also given. The test examples, which include a variable coefficient, inhomogeneous and hyperbolic-type equations, demonstrate the capability of HAM for nonlinear fPDEs.

O. Abdulaziz

I. Hashim

A. Saif

Series Solutions of Time-Fractional PDEs by Homotopy Analysis Method

This work is interested in the study of the passage of a long gravity wave above an immersed vertical barrier. The latter is placed at a right angle in the middle of the occupied fluid domain which is limited vertically by both a free surface and an impermeable horizontal bottom. We want to determine the field velocity and the local disturbances in the vicinity of the barrier. For this, we use the generalized theory of shallow water and complex variables method. For illustration, we consider a solitary wave as an emitted long wave.

Laouar Abdelhamid

Guerziz Allaoua

Numerical Simulation of the Field Velocities and Local Disturbances of a Long Gravity Wave Passing above an Immersed Vertical Barrier


                  The least-squares finite element method (LSFEM) has received increasing attention in recent years due to advantages over the Galerkin finite element method (GFEM). The method leads to a minimization problem in the   <svg style="vertical-align:-3.27605pt;width:16.1625px;" id="M1" height="15.25" version="1.1" viewBox="0 0 16.1625 15.25" width="16.1625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g><g transform="matrix(.012,-0,0,-.012,9.763,15.187)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g></svg>-norm and thus results in a symmetric and positive definite matrix, even for first-order differential equations. In addition, the method contains an implicit streamline upwinding mechanism that prevents the appearance of oscillations that are characteristic of the Galerkin method. Thus, the least-squares approach does not require explicit stabilization and the associated stabilization parameters required by the Galerkin method. A new approach, the bubble enriched least-squares finite element method (BELSFEM), is presented and compared with the classical LSFEM. The BELSFEM requires a space-time element formulation and employs bubble functions in space and time to increase the accuracy of the finite element solution without degrading computational performance. We apply the BELSFEM and classical least-squares finite element methods to benchmark problems for 1D and 2D linear transport. The accuracy and performance are compared. 

Rajeev Kumar

Brian H. Dennis

Bubble-Enriched Least-Squares Finite Element Method for Transient Advective Transport

The time evolution of the multispecies Lotka-Volterra system is investigated by the homotopy analysis method (HAM). The continuous solution for the nonlinear system is given, which provides a convenient and straightforward approach to calculate the dynamics of the system. The HAM continuous solution generated by polynomial base functions is of comparable accuracy to the purely numerical fourth-order Runge-Kutta method. The convergence theorem for the three-dimensional case is also given.

A. Sami Bataineh

M. S. M. Noorani

Series Solution of the Multispecies Lotka-Volterra Equations by Means of the Homotopy Analysis Method

The homotopy analysis method (HAM) is applied to obtain the approximate traveling wave solutions of the coupled Whitham-Broer-Kaup (WBK) equations in shallow water. Comparisons are made between the results of the proposed method and exact solutions. The results show that the homotopy analysis method is an attractive method in solving the systems of nonlinear partial differential equations.

M. M. Rashidi

D. D. Ganji

S. Dinarvand

Approximate Traveling Wave Solutions of Coupled Whitham-Broer-Kaup Shallow Water Equations by Homotopy Analysis Method

A new approach for demonstrating the global stability of ordinary differential equations is given. It is shown that if the curvature of solutions is bounded on some set, then any nonconstant orbits that remain in the set, must contain points that lie some minimum distance apart from each other. This is used to establish a negative-criterion for periodic orbits. This is extended to give a method of proving an equilibrium to be globally stable. The approach can also be used to rule out the sudden appearance of large-amplitude periodic orbits.

C. Connell McCluskey

On Using Curvature to Demonstrate Stability

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<p><em>International Journal of Differential Equations</em> publishes research on differential equations, and related integral equations, from all scientists who use differential equations as tools within their own discipline.</p>


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International Journal of Differential Equations publishes research on differential equations, and related integral equations, from all scientists who use differential equations as tools within their own discipline.

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International Journal of Differential Equations

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International Journal of Differential Equations