In this paper, two problems involving nonlinear time fractional hyperbolic partial differential equations (PDEs) and time fractional pseudo hyperbolic PDEs with nonlocal conditions are presented. Collocation technique for shifted Chebyshev of the second kind with residual power series algorithm (CTSCSK-RPSA) is the main method for solving these problems. Moreover, error analysis theory is provided in detail. Numerical solutions provided using CTSCSK-RPSA are compared with existing techniques in literature. CTSCSK-RPSA is accurate, simple and convenient method for obtaining solutions of linear and nonlinear physical and engineering problems.

In this paper, numerical technique based on monic Laguerre polynomials is proposed to obtain approximate solutions of initial value problems for systems of fractional order integro-differential equations (FIDEs). Operational fractional integral matrix is constructed. This operational matrix is applied together with the monic Laguerre Tau method to solve systems of FIDEs. This systems of FIDEs will be transformed into a system of algebraic equations which can be solved easily. Numerical results and comparisons with other methods are also presented to show the efficiency and applicability of the proposed method.

The Klein-Gordon problem (KGP) is one of the interesting models that appear in many scientific phenomena. These models are characterized by memory effects, which provide insight into complex phenomena in the fields of physics. In this regard, we propose a new robust algorithm called the confluent Bernoulli approach with residual power series scheme (CBCA-RPSS) to give an approximate solution for the fractional nonlinear KGP. The convergence, uniqueness and error analysis of the proposed method are discussed in detail. A comparison of the numerical results obtained by CBCA-RPSS with the results obtained by some well-known algorithms is presented. Numerical simulations using base errors indicate that CBCA-RPSS is an accurate and efficient technique and thus can be used to solve linear and nonlinear fractional models in physics and engineering. All the numerical results for the studied problems were obtained through implementation codes in Matlab R2017b.