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In this article, based on the operational matrix of fractional order integration, we introduce a method for the numerical solution of fractional strongly nonlinear Duffing oscillators with cubic-quintic-heptic nonlinear restoring force and then use it in some cases. For this purpose, concerning the Caputo sense, we implement the block-pulse wavelets matrix of fractional order integration. To reach this aim, we analyse the errors. The approach has been examined by some numerical examples and changes in coefficients as well as in the derivative of the equation too. It is shown that this method works well for all the parameters and order of the fractional derivative. Results indicate the precision and computational performance of the suggested algorithm.

In this work we study the existence of new periodic solutions for the well knwon class of Duffing differential equation of the form x" + cx' + a(t)x + b(t)x3 = h(t), where c is a real parameter, a(t), b(t) and h(t) are continuous T-periodic functions. Our results are proved using three different results on the averaging theory of first order.

We provide sufficient conditions for the existence of periodic solutions for the class of Duffing differential equations x ″ + c ( t ) x ′ + a ( t ) x + b ( t ) x 3 = h ( t , x ) , x'' + c(t)x' + a(t)x + b(t){x^3} = h(t,x), where the functions a(t), b(t), c(t) and h(t, x) are C 2 and T–periodic in the variable t.

We use three different results of the averaging theory of first order for studying the existence of new periodic solutions in the two Duffing differential equations ¨ y + asiny = bsint and ¨ y + ay−cy3 = bsint, where a, b and c are real parameters.

Agraïments: The research of AR is supported by the Swedish Research Council (VR Grant 2010/5905). The authors would like to thank the Göran Gustafsson Foundation UU/KTH for financial support. We study the periodic solutions of the following kind of non-autonomous Duffing differential equation y + ay − εy 3 = εh(t, y, y), with a > 0, ε a small parameter and h a C 2 function in its variables.

Agraïments: The second author is partially supported by HBP-2009-0025-PC and CAPES/MECD-DGU 015/2010 Brazil and Spain, process number BEX 4251/10-5. In this work we study the periodic solutions, their stability and bifurcation for the class of Duffing differential equation x'' + cx' +a(t)x + b(t)x3 = λh(t), where c > 0 is a constant, λ is a real parameter, a(t), b(t) and h(t) are continuous T-periodic functions. Our results are proved using the averaging method of first order.