scholarly journals Non-monotonic blow-up problems: Test problems with solutions in elementary functions, numerical integration based on non-local transformations

2018 ◽  
Vol 76 ◽  
pp. 123-129 ◽  
Author(s):  
Andrei D. Polyanin ◽  
Inna K. Shingareva
Keyword(s):  
Numerical Integration ◽  
Blow Up ◽  
Test Problems ◽  
Elementary Functions ◽  
Non Local

Estimates of blow-up time for a non-local problem modelling an Ohmic heating process

2002 ◽  
Vol 13 (3) ◽  
pp. 337-351 ◽  
Author(s):  
N. I. KAVALLARIS ◽  
C. V. NIKOLOPOULOS ◽  
D. E. TZANETIS
Keyword(s):  
Blow Up ◽  
Ohmic Heating ◽  
Numerical Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stationary Solutions ◽  
Upper And Lower Bounds ◽  
Heating Process ◽  
Non Local ◽  
Initial Boundary Value

We consider an initial boundary value problem for the non-local equation, ut = uxx+λf(u)/(∫1-1f (u)dx)2, with Robin boundary conditions. It is known that there exists a critical value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution and the solution u(x, t) blows up globally in finite time t*, while for λ < λ* there exist stationary solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t*, by using comparison methods. For f(u) = e−u, we give an asymptotic estimate: t* ∼ tu(λ−λ*)−1/2 for 0 < (λ−λ*) [Lt ] 1, where tu is a constant. A numerical estimate is obtained using a Crank-Nicolson scheme.

scholarly journals Exact solutions of the cubic Boussinesq and the coupled Higgs system

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 333-342
Author(s):  
Mahmoud Abdelrahman ◽  
Hanan Alkhidhr ◽  
Dumitru Baleanu ◽  
Mustafa Inc
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Test Problems ◽  
Explicit Solutions ◽  
Computational Results ◽  
Special Test ◽  
Elementary Functions ◽  
The Unified Method ◽  
Unified Method

We present explicit exact solutions of some evolution equations including cubic Boussinesq and coupled Higgs system by the unified method. The explicit solutions are expressed in terms of some elementary functions including trigonometric, exponential, and polynomial. The method is applied to a number of special test problems to test the strength of the method and computational results indicate the power and efficiency of the method.

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